Gorenstein toric Fano varieties [Elektronische Ressource] / Benjamin Nill

Gorenstein toric Fano varietiesBenjamin NillDissertationder Fakult˜ at fur˜ Mathematik und Physikder Eberhard-Karls-Universit˜ at Tubingen˜zur Erlangung des Grades einesDoktors der Naturwissenschaften vorgelegt2005Tag der mundlic˜ hen Qualiflkation: 22.07.2005Dekan: Prof. Dr. P. Schmid1. Berichterstatter: Prof. Dr. V. Batyrev2.h Prof. Dr. J. HausenGorenstein toric Fano varietiesBenjamin Nill(Tubingen)˜Dissertationder Fakult˜ at fur˜ Mathematik und Physikder Eberhard-Karls-Universit˜ at Tubingen˜zur Erlangung des Grades einesDoktors der Naturwissenschaften vorgelegt2005Fur˜ JuleContentsIntroduction 9Notation 151 Fans, polytopes and toric varieties 191.1 Cones and fans . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.2 The classical construction of a toric variety from a fan . . . . . . 201.3 The category of toric varieties . . . . . . . . . . . . . . . . . . . . 201.4 The class group, the Picard group and the Mori cone . . . . . . . 211.5 Polytopes and lattice points . . . . . . . . . . . . . . . . . . . . . 231.6 Big and nef Cartier divisors . . . . . . . . . . . . . . . . . . . . . 261.7 Ample Cartier divisors and projective toric varieties . . . . . . . 282 Singularities and toric Fano varieties 312.1 Resolution of singularities and discrepancy . . . . . . . . . . . . . 312.2 Singularities on toric varieties . . . . . . . . . . . . . . . . . . . . 352.3 Toric Fano varieties . . . . . . . . . . . . . . . . . . . . . . . . . .
Publié le : samedi 1 janvier 2005
Lecture(s) : 55
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Source : W210.UB.UNI-TUEBINGEN.DE/DBT/VOLLTEXTE/2005/1888/PDF/NILL.PDF
Nombre de pages : 181
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9ductiontroIn15Notation1Fans,polytopesandtoricvarieties19
1.1Conesandfans............................19
1.2Theclassicalconstructionofatoricvarietyfromafan......20
1.3Thecategoryoftoricvarieties....................20
1.4Theclassgroup,thePicardgroupandtheMoricone.......21
1.5Polytopesandlatticepoints.....................23
1.6BigandnefCartierdivisors.....................26
1.7AmpleCartierdivisorsandprojectivetoricvarieties.......28
2SingularitiesandtoricFanovarieties31
2.1Resolutionofsingularitiesanddiscrepancy.............31
2.2Singularitiesontoricvarieties....................35
2.3ToricFanovarieties..........................36
3Reflexivepolytopes41
3.1Basicproperties............................43
3.2Projectingalonglatticepointsontheboundary..........46
3.3Pairsoflatticepointsontheboundary...............52
3.4Classificationresultsinlowdimensions...............55
3.5Sharpboundsonthenumberofvertices..............58
3.6Reflexivesimplices..........................68
3.6.1Weightsystemsofsimplices.................68
3.6.2Themainresult.......................73
3.7Latticepointsinreflexivepolytopes................81
3.7.1TheEhrhartpolynomial...................81
3.7.2Boundsonthevolumeandlatticepoints..........83
3.7.3Countinglatticepointsinresidueclasses..........88
4TerminalGorensteintoricFano3-folds89
4.1Primitivecollectionsandrelations.................90
4.2Combinatoricsofquasi-smoothFanopolytopes..........94
4.2.1Definitionandbasicproperties...............94
4.2.2Projectionsofquasi-smoothFanopolytopes........97

7

8

5

6

Contsten

4.3Classificationofquasi-smoothFanopolytopes...........101
4.3.1Themaintheorem......................101
4.3.2ClassificationwhennoAS-pointsexist...........102
4.3.3ClassificationwhenAS-pointsexist.............104
4.4Tableofquasi-smoothFanopolytopes...............113

Thesetofroots119
5.1Thesetofrootsofacompletetoricvariety............121
5.2Thesetofrootsofareflexivepolytope...............130
5.3Criteriaforareductiveautomorphismgroup...........133
5.4Symmetrictoricvarieties.......................138
5.5Successivesumsoflatticepoints..................140
5.6Examples...............................143

Centrallysymmetricreflexivepolytopes147
6.1Roots.................................148
6.2Vertices................................149
6.3Classificationtheorem........................151
6.4Embeddingtheorems.........................157
6.5Latticepoints.............................160

Index

yBibliograph

AppendixA-ZusammenfassungindeutscherSprache

AppendixB-Lebenslauf

165

168

175

181

Inductiontro

InthisthesisweconcernourselveswithGorensteintoricFanovarieties,that
is,withcompletenormaltoricvarietieswhoseanticanonicaldivisorisanam-
pleCartierdivisor.Thesealgebraic-geometricobjectscorrespondtoreflexive
polytopesintroducedbyBatyrevin[Bat94].Reflexivepolytopesarelattice
polytopescontainingtheoriginintheirinteriorsuchthatthedualpolytope
alsoisalatticepolytope.ItwasshownbyBatyrevthattheassociatedvari-
etiesareambientspacesofCalabi-Yauhypersurfacesandtogetherwiththeir
dualsnaturallyyieldcandidatesformirrorsymmetrypairs.Thishasraised
alotofinterestinthisspecialclassoflatticepolytopesamongphysicistsand
mathematicians.Itisknownthatinfixeddimensiondthereonlyareafinite
numberofisomorphismclassesofd-dimensionalreflexivepolytopes.Usingtheir
computerprogramPALP[KS04a]KreuzerandSkarkesucceededinclassifying
d-dimensionalreflexivepolytopesford≤4[KS98,KS00,KS04b].Theyfound
16isomorphismclassesford=2,4319ford=3,and473800776ford=4.
Whiletherearemanypapersdevotedtothestudyandclassificationofnon-
singulartoricFanovarieties[WW82,Bat82a,Bat82b,Bat99,Sat00,Deb03,
Cas03a,Cas03b],inthesingularcasetherehasnotyetbeendonesomuch,es-
peciallyinhigherdimensions.Thiscanbeexplainedbyseveraldifficulties:First
manyalgebraic-geometricmethodslikebirationalfactorization,Riemann-Roch
orintersectiontheorycannotsimplybeapplied,especiallysincethereneednot
existacrepanttoricresolution.Secondmostconvex-geometricproofsreliedon
theverticesofafacetformingalatticebasis,afactwhichisnolongertrue
forreflexivepolytopes,wherefacetscanevencontainlatticepointsintheirin-
terior.Thirdthehugenumberofreflexivepolytopescausesanyclassification
approachtodependheavilyoncomputercalculations,henceoftenwedonot
getmathematicallysatisfyingproofsevenwhenrestrictingtolowdimensions.
Theaimofthisthesisistogiveafirstsystematicmathematicalinvestiga-
tionofGorensteintoricFanovarietiesbythorouglyexaminingthecombinatorial
andgeometricpropertiesoftheirconvex-geometriccounterparts,thatis,reflex-
ivepolytopes.Wewouldliketogeneralizeusefultoolsandtheoremspreviously
onlyknowntoholdfornonsingulartoricFanovarietiestothecaseofmildsingu-
laritiesandtoproveclassificationtheoremsinimportantcasesandinarbitrary
dimension.Moreoverweareinterestedinfindingconstraintsonthecombina-
toricsofreflexivepolytopesandconjecturesandsharpboundsoninvariants
thatcanexplaininterestingobservationsmadeinthelargecomputerdata.
Aswewillseemostoftheseaimsareactuallynotoutofreach,forinstance
itwillunexpectedlyturnoutthatQ-factorialGorensteintoricFanovarieties
areinmanyaspectsnearlyasbenignasnonsingulartoricFanovarieties.More-
overbythesegeneralizationswithastrongfocusoncombinatoricsevenresults
previouslyalreadyproveninthenonsingularcasebecomemoretransparent.
9

10

ductiontroIn

Aboveallthisworkprovidesusefultools,severalconjecturestoproveinthe
futureandmanyresultsthatshowreflexivepolytopestobetrulyinteresting
objects-notonlyfromaphysicist’sbutalsofromapuremathematician’spoint
view.ofThisthesisisorganizedinsixchapters.Anymajorchapter(3-6)startswith
anintroductorysection,inwhichalsoanexplicitlistofthemostimportantnew
resultsiscontained.Furthermorethereaderwillfindrightafterthisintroduc-
tionasummaryofnotationandattheendofthisthesisanindexaswellasa
.ybibliographecomprehensivThefirsttwochapterscoverthenotionsthatarebasicforthiswork.
Chapter1fixesthemainnotationandgivesasurveyofimportantresults
.geometrytoricfromChapter2givesanexpositionoftoricFanovarietiesandclassesofsingu-
laritiesthatappearnaturallywhentryingtodesingularizetoricFanovarieties.
Herewesetupthedictionaryofconvex-geometricandalgebraic-geometricno-
tions:FanopolytopescorrespondtotoricFanovarieties,smoothFanopolytopes
tononsingulartoricFanovarietiesandcanonical(respectivelyterminal)Fano
polytopestotoricFanovarietieswithcanonical(respectivelyterminal)singu-
larities.MoreoversimplicialFanopolytopesareassociatedtoQ-factorialtoric
arieties.vanoFChapter3istheheartofthisthesis.Herethemainobjectsofstudyarein-
troduced:ReflexivepolytopescorrespondingtoGorensteintoricFanovarieties.
Atthebeginningtwoelementarytechnicaltoolsareinvestigatedandgener-
alizedsingularthattorichavFeanovalreadyarietiesbeen[Bat99used,toSat00,successfullyDeb03,invCas03bestigate].Theandfirstclassifyone,thatnon-
isespeciallyusefulinlowerdimensions,istheprojectionmap.Weprovesome
generalfactsaboutprojectionsofreflexivepolytopes(Prop.3.2.2),therebywe
canrelatethepropertiesofGorensteintoricFanovarietiestothatoflower-
dimensionaltoricFanovarieties.Asanapplicationwepresentgeneralizations
tomildsingularitiesofanalgebraic-geometricresultduetoBatyrev[Bat99]
statingthattheanticanonicalclassofatorus-invariantprimedivisorofanon-
singulartoricFanovarietyisalwaysnumericallyeffective(Cor.3.2.7,Prop.
3.2.9).Moreoverwegetasatrivialcorollary(Lemma3.5.6)thatalatticepoint
ontheboundarythathaslatticedistanceonefromafacetFhastobecon-
tainedinafacetFintersectingFinacodimensiontwoface.Previouslythis
observationcouldbeprovenbyDebarrein[Deb03]onlyinthecaseofasmooth
FanopolytopebyusingalatticebasisamongtheverticesofF.
Thesecondimportanttoolisthenotionofprimitivecollectionsandrela-
tions.ItwasintroducedbyBatyrevin[Bat91]tocompletelydescribesmooth
Fanopolytopesandhasbeenessentialforhisclassificationofnonsingulartoric
Fano4-folds[Bat99].Ingeneralthistoolisnotapplicableforreflexivepoly-
topes,sinceitusestheexistenceoflatticebasesamongthevertices.Thespecial
caseofaprimitivecollectionoflengthtwocorrespondstoapairoflatticepoints
ontheboundarythatdonotlieinacommonfacet.Thissituationisextremely
importantandwasinvestigatedbyCasagrandein[Cas03a]toprovesomestrong
restrictionsonsmoothFanopolytopes.Nowtheauthorhasbeenabletosuit-
ablygeneralizethisnotiontoreflexivepolytopes(Prop.3.3.1)andapplyit
successfullytoshowthatthesamerestrictionsalsoholdforsimplicialreflexive
polytopes.Inalgebraic-geometriclanguagethecorrespondingstatementreads

ductiontroIn

11

asfollows:thePicardnumberofaQ-factorialGorensteintoricFanovariety
exceedsthePicardnumberofatorus-invariantprimedivisoratmostbythree
(Cor.3.5.17).Asanotherapplicationweprovethatanypairofverticesofa
simplicialreflexivepolytopecanbeconnectedbyatmostthreeedges,wherethe
casethatlessthanthreeedgesdonotsufficecanonlyoccurforacentrallysym-
metricpairofvertices(Cor.3.3.2).Ingraph-theoreticlanguagethisyieldsthat
thediameteroftheedge-graphofasimplicialreflexivepolytopeisatmostthree.
Fromthisweeasilygetthatcertaincombinatorialtypescannotberealizedas
reflexivepolytopes(Cor.3.3.4).
Themainpartofthethirdchapterdealswithupperboundsonthevolume
andthenumberoflatticepointsandverticesofareflexivepolytope.
Wed/2stateinConjecture3.5.2thatad-dimensionalreflexivepolytopehasat
most6vertices,whereequalityholdsonlyforonespecialeven-dimensional
reflexivepolytope.Wealsopresentsomepreliminarycoarseboundsdepending
onthecombinatoricsofthefacets(Prop.3.5.5).Inthelastchaptertheconjec-
turewillbeproventoholdforcentrallysymmetricsimplereflexivepolytopes.
Hereinthethirdchapterthemainfocusisonthesimplicialcase:Weextend
thelong-standingconjectureofBatyrevonthemaximalnumberofverticesof
smoothFanopolytopestosimplicialreflexivepolytopes(Conj.3.5.7):Itstates
thatad-dimensionalsimplicialreflexivepolytopehasatmost3dvertices,ifd
iseven,and3d−1vertices,otherwise.Wehavealsoincludedinthisconjecture
theextrastatementthatthenumberof3dverticesshouldonlybeobtainedin
oneuniquecase.Nowusingthepreviouslydescribedtoolswegiveaproofin
thecaseofacentrallysymmetricpairofverticesofthedualpolytope(Thm.
3.5.11).CasagrandeBasedhasboneentheseabletoresultssuccessfullythatwereprovepublishedConjectureinthe3.5.7inpreprint[Cas04].[Nil04a]
Inlowdimensionsthemaximalnumberoflatticepointsofareflexivepoly-
topeisachievedonlybysomereflexivesimplices.Henceitmakessensetorestrict
tothissituationinordertofindagoodgeneralupperbound.Apossibleway
todothisistoproveasharpupperboundonthevolumeofareflexivesim-
plex.ThishasbeenachievedbytheauthorinTheorem3.7.13.Tothisendwe
describeinsection3.6thecommonapproachofBatyrev[Bat94]andConrads
[Con02]todeterminelatticesimplicesbysocalledweightsystems.Inthecase
ofreflexivesimplicestheycorrespondtounitfractionssumminguptoone,e.g.,
21+31+61=1.Nowtheprooffollowsfromupperboundsonthedenominatorsof
theseunitfractions(Prop.3.6.29).Moreoverusingthisapproachwecanprove
inThm.3.7.19anobservationofHaaseandMelnikov[HM04]statingthatthere
isauniqued-dimensionalreflexivesimplexwiththemaximalnumberoflattice
pointsonanedge,namely2yd−1−1,forthesequence(yn)definedasy0=2
andyn=1+y0∙∙∙yn−1.
Soweseethatinthissituationconvex-geometricquestionsnaturallycor-
respondtonon-trivialproblemsinelementarynumbertheory.Howeverthese
resultsmayalsobetranslatedintoalgebraicgeometry:Forinstancethebound
onthevolumeyieldsasharpupperboundontheanticanonicaldegreeofa
d-dimensionalGorensteintoricFanovarietyXwithclassnumberone,inpar-
ticularonweightedprojectivespaceswithGorensteinsingularities(Thm.3.7.7):
Ford=2,respectivelyd=3,thedegreeofXisatmost9,respectively72;for
d≥4thedegreeofXisatmost2(yd−1−1)2,andequalityholdsonlyforone
specialvariety.Thisboundford=3(calledtheFano-Iskovskikhconjecture)
hasrecentlybeenprovenbyProkhorov[Pro04]forthree-dimensionalGoren-

12

ductiontroIn

steinFanovarietieswithcanonicalsingularities.Nowthepreviousresultyields
toenoughhigherevidencedimensions.tomotivThisateashowsagaingeneralizationthatoftoricthevFarietiesano-Iskoarevskikhfertileconjecturetesting
groundsforconjecturesonmoregeneralvarieties.
Wealsodescribeasanotherusefultool”countingmoduloanaturalnumber”
asintroducedbyBatyrevin[Bat82a].Thissimplemethodcanbesuccessfully
appliedtoproveasharpboundonthenumberoflatticepointsinterminal
reflexivepolytopes(Cor.3.7.23)andwillalsobeusedinthelastchapter.
Chapter4givesafirstapplicationofthetechniquesdevelopedintheprevi-
ouschapter:acomplete,explicit,computer-independentclassificationofthree-
dimensionalGorensteintoricFanovarietieswithterminalsingularities.There
areprecisely100isomorphismclasses,correspondingtosocalledquasi-smooth
Fanopolytopes(Thm.4.3.2).Themainideaoftheproofistoshowthatin
mostcasesthereexistsavertexthatiscentrallysymmetrictosomeothervertex
andatthesametimethesumoftwoothervertices.Thenweusetheresult
(Prop.4.2.17)thatthecorrespondingquasi-smoothFanopolytopeisuniquely
determinedbyasmallsetofspecialrelationsamongitsvertices,thesearecalled
quasi-primitiverelations.Tothisendweincludeageneraldiscussionofvarious
notionsofprimitivecollectionsandrelationsatthebeginningofthischapter.
sociatedChaptertoa5iscompleteconcernedfan,withcalledasptheecialsetofsetroofots.latticeForptheointsfanthatofcannormalsbeas-of
areflexivepolytopethesetofrootsispreciselythesetoflatticepointsinthe
interioroffacets.Rootsareimportantinordertodescribetheautomorphism
groupoftheassociatedtoricvariety,inparticularitsdimension.Centrallysym-
metricrootsarecalledsemisimpleroots.Theautomorphismgroupisreductive
ifdescribandeonlytheifsetanofyrorootots:isThesemisimple.morealgebraicThereonearetduewotodifferenCoxt[Cox95approac]useshestheto
notionofthehomogeneouscoordinateringandcanbeappliedtocompletetoric
varieties.ThemoregeometriconeisduetoBrunsandGubeladze[BG99]and
directlyconcernedwithlatticepolytopes.Theauthorfirstappliedtothesetof
rootstheresultsinthethirdchapteraboutpairsoflatticepointsinreflexive
polytopes,hereitturnedoutBrunsandGubeladzehadalreadyderivedsimilar
observations.Latertheauthorgaveageneralizationtocompletetoricvarieties
usingtheapproachofCox.Themainideaistointroducesocalledfacetbases
androotbasesthatparametrizethesetoffacetscontainingrootsandthesetof
semisimplerootsinageometricallyconvenientway(Def.5.1.6,Prop.5.1.22).
Asanapplicationofthisnotionwederivethatacompletetoricvariety
isalreadyaproductofprojectivespaces,ifthesetofsemisimplerootsspan
thevectorspace(Prop.5.1.19).Inthecaseofareflexivepolytopeweeven
getthattheintersectionofthereflexivepolytopewiththespacespannedby
allsemisimplerootsisagainareflexivepolytopeassociatedtoaproductof
forproP3jectivtheespacesgeneral(Thm.phenomenon5.2.12).The5.1.18figurethatonthetheconvtitleexofhullthisofwallorksemisimpleillustrates
rootsisalsoalwaysareflexivepolytope.Furtherweobtainthatad-dimensional
reflexivepolytopehasatmost2dfacetscontainingroots,whereequalityimplies
thetoricvarietytobeaproductofprojectivelines(Cor.5.2.4).Asoneofthe
mainresultsweprovethatthereductiveautomorphismgroupofad-dimensional
completetoricvarietythatisnotaproductofprojectivespaceshasatmost
dimension2ford=2,respectivelyd2−2d+4ford≥3(Thm.5.1.25).This

ductiontroIn

13

sharpboundyieldsanexplanationforobservationsonthenumberofrootsof
reflexivepolytopesduetoKreuzerandtheauthorinthedatabase[KS04b].
Oneofthemainmotivationtostudythesetofrootswasgivenbythegoalto
findcombinatorialcriteriafortheautomorphismgrouptobereductive.There
isawell-knownresultduetoMatsushimathattheautomorphismgroupofa
nonsingularFanovarietyisreductive,ifthevarietyadmitsanEinstein-K¨ahler
metric.In[BS99]BatyrevandSelivanovashowedthatsuchametricexists,if
thereflexivepolytope,whosefanofnormalsisassociatedtothetoricvariety,is
symmetric,i.e.,thegroupoflinearautomorphismsofthepolytopehasnonon-
zerofixpoint.Hencetheygotthatsymmetryimpliesanyroottobesemisimple,
andtheyaskedforapurelyconvex-geometricproofofthiscorollary.Moreover
theexistenceofanEinstein-K¨ahlermetricimpliesthebarycenterofthisreflexive
polytopetobezero,thereverseimplicationcouldonlybeprovenveryrecently
in[WZ04]byWangandZhu.Howeveratthebeginningofthisresearchitwas
notclearthateveninthecaseofanonsingulartoricFanovarietythevanishing
ofthebarycenterimpliesallrootstobesemisimple,yetthiswasobservedby
BatyrevandKreuzerevenforanyreflexivepolytopeuptodimensionfourinthe
computerdatabaseandtheyconjecturedthatthisshouldholdinanydimension.
Nowthischaptercontainspurelyconvex-geometricproofsoftheseconjec-
turesandimplicationsingreatestpossiblegenerality,aswellassomeothercom-
binatorialcriteriathataresufficientfortheautomorphismgrouptobereductive
(Thm.5.3.1).Weillustratetheseresultsbyseveralexamplesandclassifyall
three-dimensionalsymmetricreflexivepolytopes(Thm.5.4.5).Moreoverwe
show,howthesecriteriarelatetotheapproachofKreuzer[Kre03a,Kre03b]to
investigatesumsoflatticepointsinmultiplesofareflexivepolytopebyavariant
olynomial.pErharttheofChapter6givesasafinalapplicationoftheresultsachievedsomeinsight
incentrallysymmetricreflexivepolytopes.Amainresultstatesthattheunit
latticecube[−1,1]distheoneandonlyd-dimensionalcentrallysymmetricre-
flexivepolytopewiththemaximalnumberoflatticepoints(Thm.6.5.1).For
theproofweusethefactthattheunitlatticecubeisalsotheonlysuchpolytope
withthemaximalnumberof2droots(Thm.6.1.1).
Inthecaseofasimplecentrallysymmetricreflexivepolytopewecanprove
thegeneralConjecture3.5.2onthemaximalnumberofvertices(Thm.6.2.2).
Thisisactuallyanapplicationoftheothermainresultofthischapter:acom-
pleteclassificationofarbitrary-dimensionalsimplicialreflexivepolytopeshaving
acentrallysymmetricpairoffacets(Thm.6.3.1,Cor.6.3.3).Thisisaunifying
generalizationofresultsofEwald[Ewa88]andhisstudents[Wag95,Wir97];the
proofshereareconsiderablysimplersinceweusedualbasesratherthancal-
culationswithdeterminants.Applyingtod≤5yields4,5,15,20isomorphism
classesofthesed-dimensionalpolytopes(Thm.6.3.12).Asacorollaryweget
thatanyd-dimensionalsimplicialreflexivepolytopewithacentrallysymmetric
pairoffacetscanbeembeddedintheunitlatticecube[−1,1]d,whilethedual
polytopecanbeembeddedind/2[−1,1]d.Moreoverweproveageneralre-
sultonembeddingareflexivepolytopeintoamultipleoftheunitlatticecube
undersomemildassumptions(Thm.6.4.4).Finallywedeterminethemaximal
numberoflatticepoints,namely2d2+1,asimplicialreflexivepolytopewith
acentrallysymmetricpairoffacetscanhave.Againthereisonlyonesuch
polytopewiththisnumberofvertices(Thm.6.5.3).

14

twledgmenknoAc

Inductiontro

VictorFirstV.ofBatallIyrevwanfortptoosingexpressproblems,myhisgratitudeadvicetomandythesisencouragemenadvisort.Professor
IamthankfultoProfessorKlausAltmannforkindhospitalityattheFU
Berlinandthepossibilityofgivingatalk.IwouldalsoliketothanktheDFG
Forschungsschwerpunkt”GlobaleMethodeninderkomplexenGeometrie”for
ort.suppfinancialIdiscussionsamgratefulresolvingtoinProfessorExampleChristian5.1.26,hisHaaseinforvitationhistogentheuineinAlgebraicterest,fruitfulGeom-
etrythere.IseminaramatthankfulDuketoUnivtheersitDukyeandUnivforersittheywforarmfinancialhospitalitsuppyIort.expItiseriencedalso
apleasuretothankMilenaHeringforherinvitationtothestudentAlgebraic
GeometryseminarattheUniversityofMichiganandforthenicetimeIhad
there.forFhisurtherkindIwsuppouldortlikewithtotheexpressmyclassificationgratitudedatatoandProfessorhiseffortsMaximillianwiththeKreuzercom-
puterdatabasepackage[KS04bP]ALPhas,bandeenforofmanenormousymotivuseatingtome,examplessinceitandgivesobservations.mathematiciansThe
thepossibilitytotestandfindmanyinterestingconjectures.FurthermorePro-
fessorKreuzerinitiatedandparticipatedinstudyingthepropertiesofsuccessive
sumsoflatticepointsinreflexivepolytopesasdocumentedinthemanuscripts
[Kre03a,Kre03b]thatwerekindlygiventomydisposal.
ThanksalsogotoProfessorAnnetteA’Campo-Neuenforgivingreference
toXiaoh[B¨uauh96Zh],uforProfessorgivingG¨unterreferenceEwaldtofor[WZ04giving],andreferenceProfessortoOlivier[Wir97],DebarreProfessorfor
anexpositionoftheproofof[Deb03,Theorem8].MoreoverIwouldliketo
thankProfessorG¨unterZieglerforgivingsmartanswerstostupidquestions.
LastbutnotleastIwishtothankMarkBlume,TobiasD¨ohler,Andreas
forHermanntheenandtirelystimJulianeulatingRostenforprovironment.of-readingTheandaxiomtheofcResearchoicehwasGroupnotharmedAlgebra
duringthemakingofthisthesis.

Iamforeverindebtedtomyfamilyforunbreakablefaith,unbelievablesup-
portanduncountablesignsoflove.

Notation

GeneralN={0,1,2,...};0isanaturalnumberbutnotpositive
αkcalculatingmodulok(p.88)
x(x)thegreatest(lowest)integerlower(greater)orequaltox∈R
...,aˆi,...the”hat”-symbolmeansthattheithentryaiisleftout
N,M,∙,∙duallatticesNandMwithpairing∙,∙(p.19)
NR,MRassociatedrealvectorspaces
dusuallythedimensionofN,M,NR,MR
lin,affthelinearspan,respectivelytheaffinespan(p.19)
pos,convthepositivehull,respectivelytheconvexhull(p.19)
<A>ZthesetofintegerlinearcombinationsofelementsinA⊆M
ansFusuallya(complete)fan(ofteninN)(p.19)
(k)thesetofk-dimensionalconesof
supp()thesupportof(p.21)
X(N,)thetoricvarietyassociatedtolatticeNandfan(p.20)
cone(full-dimensional)ausuallyσσ∨thedualconeofσ(p.19)
τusuallyaray,i.e.,aone-dimensionalconein
Vτthetorus-invariantprimedivisorassociatedtorayτ(p.21)
vτtheuniqueprimitivelatticepointonrayτ(p.22)
SF(N,)thesetofpiecewiselinearfunctions(p.22)
hoftenanelementinSF(N,)
DhtheCartierdivisorassociatedtoh(p.22)
PhthepolytopeofglobalsectionsofDh(p.26)
Gσ,Gsetofprimitivelatticepointsonraysofσ,resp.(p.35,37)
Q,PQ=conv(G),P=Q∗(p.37,37)
v=ΣPv(p.49)
fanassociatedtosomerayτ(p.49)
Importantpolytopesandvarieties
Edthed-dimensionalreflexivesimplexwithX(M,ΣEd)=∼Pd(p.126)
Zdthed-dimensionalstandardlatticezonotope(p.58)
S3thetwo-dimensionaldelPezzosurfaceX(M,ΣZ2)(p.58)
Fdthed-dimensionaldelPezzopolytope(p.153)
Wdthed-dimensionaldelPezzovarietyX(M,ΣFd)(p.153)
F˜dthed-dimensionalpseudodelPezzopolytope(p.153)
W˜dthed-dimensionalpseudodelPezzovarietyX(M,ΣF˜d)(p.153)
Ddd-dimensionalsimplicialreflexivepolytopewith2dvertices(p.155)
Dd=X(M,ΣDd)(p.155)
15

16

Notation

esolytopPP∗usuallyad-dimensionalpolytope(ofteninMR)
PthedualpolytopeofP(p.24)
NPthefanofnormalsofP(p.24)
ΣPthefanspannedbythefacesofP
XPthetoricvarietyassociatedtothenormalfanofP
bPthebarycenterofP(p.25)
wbPtheweightedbarycenterofP(p.140)
ePtheEhrhartpolynomialofP(p.25)
sPthelatticepointsumpolynomialofP(p.140)
rPtherootsumpolynomialofP(p.143)
vol(P)thevolumeofP,ifPisfull-dimensional
W(P)thegraphoflatticepointsontheboundaryofP(p.53)
AutM(P)automorphismsofMleavingP⊆MRinvariant
∂PtheboundaryofP
intPtheinteriorofP,ifPisfull-dimensional
G≤PGisafaceofP
V(P)thesetofverticesofP
F(P)thesetoffacetsofP
FusuallyafacetofP
ηFtheuniqueinnernormalofF,i.e.,ζF,F=−1(p.24)
ζFtheuniqueprimitiveinnernormalofF(p.25)
νF=−ηF
relintFtherelativeinteriorofF
rvol(F)therelativevolumeofF(p.25)
detaff(F)determinantofaffinesublatticegeneratedbyF
v,w,x,yusuallylatticepoints
[x,y]=conv(x,y),similarly]x,y],]x,y[
v∼wv,warecontainedinacommonfacet
vwvisawayfromwforv,wlatticepointson∂P(p.47)
st(v)thestarsetofvforv∈∂P(p.47)
∂vthelinkofvforv∈∂P(p.47)
πvtheprojectionmapalongalatticepointv(p.48)
ιvtheinversemapofπvfromPvontost(v)(p.48)
Mv,PvMv=M/Zv,Pv=πv(P)
Xv=X(Mv,v)(p.49)
z(v,w)latticepointin∂P,ifv∼wandv+w=0(p.52)
Quasi-smoothFanopolytopes
PaprimitivecollectionofP(p.91)
σ(P)thesumoftheelementsinP(p.91)
deg(P)thedegreeofP(p.92)
LR(P)groupoflinearrelationsoflatticepointsinP(p.93)
πi,Pi,Mi,ιi=πvi,πvi(P),Mvi,ιviforvertexvi(p.97)
∂M(vi)=∂(vi)∩M(p.97)
deg(vi)=|∂M(vi)|,thedegreeofthevertexvi(p.97)
mi(v),bi(v)numbersassociatedtoverticesvi,v(p.104)
n=|V(P)|forPquasi-smoothFanopolytope(p.113)
fi,pnumberofi-dim.faces,parallelogramfacets(p.113)
λ(P),M(P)λ(P)=n−3−ρX,rankofthematrixM(P)(p.113)

Notation

arietiesvoricTXusuallya(toric)varietyofdimensiond(p.20)
Aut(X)automorphismgroup,Aut◦(X)isconnectedcomponent
KX,−KXcanonical,respectivelyanticanonical,divisorofX
deg(X)=(−KX)d,the(anticanonical)degreeofX
Cl(X)TheclassgroupofX(p.21)
≡linearequivalenceofdivisors(p.21)
Pic(X)thePicardgroupofX(p.22)
ρXthePicardnumberofX(p.22)
NE(X)theMoriconeofX(p.23)
discr(X)discrepancyofX(p.34)
jXtheGorensteinindexofX(p.33)
E(1):=Exc(f)theexceptionallocusofabirationalmorphismf
Etheunionoftheexceptionaldivisors
X∗the”dual”toricvariety(p.84)
systemsteighWHerm(d,λ)Hermitenormalformmatricesofsized,detλ(p.69)
Qhereaweightsystem,i.e.,anelementofQd>+10(p.68)
QredthereductionoftheweightsystemQ(p.68)
|Q|thetotalweightoftheweightsystemQ(p.68)
λQthefactoroftheweightsystemQ(p.68)
mQaninvariantoftheweightsystemQ(p.70)
P(Q)weightedprojectivespacewithweightsystemQ
QPweightsystemassociatedtolatticepolytopeP(p.69)
λPthefactoroftheweightsystemQP(p.69)
MPthelatticegeneratedbytheverticesofP
PQsimplexassociatedtoreducedweightsystemQ(p.69)
SQpolytopeassociatedtoreducedweightsystemQ(p.72)
QdSylvesterweightsystemoflengthd(p.73)
QdenlargedSylvesterweightsystemoflengthd(p.73)
ynSylvestersequence2,3,7,43,...(p.73)
tn=yn−1=y0∙∙∙yn−1,sequence1,2,6,42,...(p.73)
otsRoRsetofrootsoffanorreflexivepolytope(p.122,130)
Ssetofsemisimpleroots(p.122,130)
U=R\S,setofunipotentroots
τm,xmassociatedray,resp.monomial,toarootm(p.124)
S1={m∈S:τmnotassociatedtosomeunipotentroot}
S2=S\S1
v⊥wvandwareorthogonal(p.123,132)
p(v,w)rootassociatedtorootsv,w(p.124)
v≡wvandwareequivalentsemisimpleroots(p.124)
FvfacetcontainingrootvofareflexivepolytopeP
ηv=ηFvforareflexivepolytope,ingeneralseep.122
Softenthehomogeneouscoordinatering(p.123)
Y,Mindeterminates,respectivelymonomials,inS(p.124)
YiequivalenceclassofindeterminatesinS(p.125)
p,q,r,snumberofspecialequivalenceclassesYi(p.125)

17

Chapter1

Fans,polytopesandtoric
arietiesv

ductiontroInThemainpurposeofthischapteristofixthenotationandtogiveanoverview
ofthemostimportantresultsabouttoricvarietiesandlatticepolytopes.All
toricvarietiesarenormal,butmaybesingular.Proofsareusuallyleftout,since
theycanbeeasilyfoundintheliterature,i.e.,[Ewa96],[Ful93]and[Oda88].

fansandCones1.1LetN=∼Zdbead-dimensionallatticeandM=HomZ(N,Z)∼=Zdthedual
latticewith∙,∙thenondegeneratesymmetricpairing.Asusual,NQ=N⊗Z
Q=∼QdandMQ=M⊗ZQ=∼Qd(respectivelyNRandMR)willdenotethe
rational(respectivelyreal)scalarextensions.
ThroughoutthisworktherolesofNandMareinterchangeable.
spanF,orofaS.subsetWeSdenoteofNRbyletposlin((SS)),theresp.positiveaff(S),hulbleoftheS,linei.e.,arthespanset,ofresp.positivaffinee
linearcombinations,andbyconv(S)=pos(S)∩aff(S)theconvexhullofS.
Apolyhedralconeσ⊆NRisthepositivehullpos(G)offinitelymanypoints
GinNR.IfadditionallythegeneratorsGarelatticepoints,i.e.,G⊆N,then
σifisitdocalledesnotapconolyhetaindralalatticlineareconesubspace,.Finallyi.e.,0a∈coneNisσanisapcalledexofstrσ.onglyconvex,
Wedefinefurtherthedualconeσ∨={u∈MR:u,v≥0∀v∈σ}.Itis
fully(d-)dimensionalifandonlyifσisstronglyconvex.Ifσisalatticecone,
thenalsoσ∨isalatticecone(FarkasLemma).
Afaceofaconeσistheintersectionofσwithanaffinehyperplanesuch
thatσiscontainedinoneofthetwohalfspaces.Adefiningvectorforsuch
anhyperplaneiscalledadefiningouter,respectivelyinner,normalforthat
face.Thenormalofacodimensiononefacecanbeuniquelydefinedbysome
condition.normalizingAfanisheredefinedtobeacollectionoffinitelymanystronglyconvex
polyhedrallatticeconessuchthatanyfaceofaconeinisalsoanelementof
19

20

Chapter1.Fans,polytopesandtoricvarieties

andtheintersectionoftwoconesinisafaceineach.Fork∈Nwelet
(k)denotethesetofk-dimensionalconesin.
1.2Theclassicalconstructionofatoricvariety
fanafromLetconeσisbeaaconefinitelyinafangenerated.ThensaturatedthesetSadditivσ=eM∩σ∨submonoidoflatticeofMpointsgeneratinginitsMdual,
i.e.,lattice<pSσoin>tZ=thatM.canbHereewrittensaturationasaforarationalpsubmonoidositiveWcomofMbinationmeansofthatelemenantsy
inWNo,wi.e.,waelatticedefineptheointinfinitelypos(W)∩generatedMQ,Cmust-AlgebraalreadyAbe:=conC[Stained],inandW.the
σσvarietycorrespassoondingciatedaffinetoscσ.hemeWhenUσ:=loSpokingecCat[Sσits].Uclosedσispoincalledts,thei.e.,affinethemax-toric
imalspectrumofAσ,theHilbertscherNullstellensatzimpliesSpmC[Sσ]=
φ(Homm)C∙−φal(gm.()C}[.Sσ],C)=Hommon(Sσ,C)={φ:Sσ→C,φ(0)=1,φ(m+m)=
isanA{0}algebr=Caic[M]torusis,i.e.,isomorphictheclosedtothepoinringtsofofTlaurenaretpSpmColynomials[M]=andHomTN:=(MU,{0C})
=HomZ(M,C∗)=N⊗ZC∗=∼(C∗)d.ThecharNacterse(m)∈A{0}ofmonthetorus
arejustthemonomialsform∈M.
τ=Ifστ∩≤u⊥σ.isEspafaceeciallyofσthe,thennaturalonemaphasAUττ=→(UAσσ,)e(onu)theforsomeclosedup∈oinStsσgivwithen
byφ→φ|Sσ,isanopenimmersionastheelementaryopensetDUσ(e(u)).The
embeddingofTNiscalledthebigtorus.
TN×ThereUσ→isUaσ,(naturalt,φ)→torust∙φpactionointofwise,TNgivinducedenonbythetheclosedmapApσoin→tsTofNU⊗σAbσy,
e(u)→e(u)⊗e(u).ThisextendstheactionofthebigtorusTonitself.
FinallywedefinethetoricvarietyX()=X(N,)assoNciatedtothefan
σ1,σas2∈thedisjoinonetgluesunion(asofscthehemes)affinealongtoricthevinarietiesUσUσund(σUσ∈m),utuallywhereopforen
21subvAnarietyalternativUσ1∩σe2.constructionofatoricvarietyXasacategoricalquotientwas
ifgivandenbyonlyCoifxaniny[Coconex95in],theherefanXcandefiningevenXbeisadescribsimplex.edasageometricquotient

1.3Thecategoryoftoricvarieties
Letbeafan.ThenthepreviouslyconstructedX()isavarietyinthesense
of[Har77,II.4.10],i.e.,anintegralseparatedschemeoffinitetypeoverC,X()
isnormalandequippedwithanaturaltorusactionbyTNextendingitsinherent
diagonalaction.OfcourseX()isadisjointunionoftheorbitsunderthetorus
action,thiscanbereadoffexplicitlyfromthefan,see[Ful93,3.1].Everycone
σ∈correspondsexactlytoadim(σ)-codimensionaltorusorbitclosureVσin
).(XWegenerallydefineatoricvarietyXasanormalirreduciblealgebraicvariety
overCwithanopenembeddedalgebraictorusT=(C∗)dactingonXin
extensionofitsownaction.

1.4.TheclassgroupandthePicardgroup

21

Togetacategoryoftoricvarietiesoneneedsasuitablenotionofamorphism:
SoletXandYbetwotoricvarietieswithitsembeddedtorusesTXandTY.
Thenamorphismofvarietiesf:Y→Xiscalledamorphismoftoricvarieties,
ifadditionallyamorphismofalgebraicgroupsf:TY→TXexists,sothatf
isequivariantwithrespecttof,i.e.,f(t∙y)=f(t)∙f(y)forallt∈TYand
.Yy∈NowagainletbeafanwiththelatticeN,andalsoafanwiththe
latticeN.AZ-linearmapφ:N→Niscalledamapoffans,iftheimageof
anyconeinunderφ⊗ZRismappedinaconein.Thenonecanconstruct
amorphismoftoricvarietiesφ∗:X(N,)→X(N,)thatisequivariant
withrespecttoφ∗|TN=φ⊗1:TN=N⊗ZC∗→TN=N⊗ZC∗.
Thisgivesanaturalmapfromthecategoryoffanstothecategoryoftoricva-
rieties.Onehasthetheoremthatthisisacovariantisomorphismofcategories.
Underthiscorrespondenceanaffinetoricvarietycorrespondstothefanofthe
facesofastronglyconvexpolyhedrallatticecone.Alsopropertoricvarieties
correspondtocompletefans.Acompletefanisafansuchthatitssupport
isthewholespaceNR,whereitssupportsupp()isdefinedastheunionofall
conescontainedin.
In[Oda88,Prop.1.33]itisexplained,whenanequivariantmorphismbe-
tweentoricvarietiescorrespondstoatoricfibre-bundle;inparticularproducts
oftoricvarietiescorrespondtoproductsoftheassociatedfans.Inthesame
wayatoricblow-up,i.e.,ablow-upofatoricvarietyalongatorus-invariant
subvariety,isassociatedtoafanthatisastarsubdivisionofthecorresponding
cone,thiscanbefoundin[Oda88,Prop.1.26].

1.4Theclassgroup,thePicardgroupandthe
coneMoriAstypicalinthecaseoftoricvarietiestheirinvariantscanbeextractedfrom
thefan.LetX=X(N,).
Ak-cycleisdefinedtobeanelementofthefreeabeliangrouponthek-
dimensionalirreducibleclosedsubvarietiesofX.DefinetheChowgroupAk(X)
ofXasthequotientofthegroupofk-cyclesmodulorationalequivalence(see
[Har77,A.1]).ThenAk(X)isgeneratedbytheclassesoforbitclosuresVσfor
σ∈(d−k).
Let’slookattheclassgroupCl(X)=Ad−1ofX,i.e.,thegroupofWeil
divisorsmodulolinearequivalence≡.Anytorus-invariantprimedivisorVτcor-
respondsexactlytoonerayτ∈(1),i.e.,aone-dimensionalconein.These
divisorsgeneratethegroupoftorus-invariantWeildivisorsTNDiv(X).Further
atorus-invariantprincipalWeildivisorisexactlygivenbydiv(e(u))foru∈M.
Wehavethefollowingrightexactsequence,whichisexact,ifeveryelementof
NRisalinearcombinationofelementsinsupp(),i.e.,lin(supp())=NR:
0→M→TNDiv(X)=ZVτ→Cl(X)→0.(1.1)
(1)∈τInparticularCl(X)isafinitelygeneratedabeliangroup,itsrankiscalledthe
classnumberofX.WhentheraysofspanNR,wehaverank(Cl(X))=r−d.
Cl(X)isingeneralnottorsionfree,evenifXisproper.

22

Chapter1.Fans,polytopesandtoricvarieties

ThecalculationofthePicardgroupPic(X),i.e.,thegroupofCartierdivisors
modulolinearequivalence,isbasedonthefollowingnotion:LetSF(N,)
denotethesetofallfunctionsh:supp()→RbeingZ-valuedonN∩supp()
andlinearonsupp(),i.e.,foreveryσ∈thereexistslσ∈Mwithh|σ=lσ|σ.
LetTNCDiv(X)denotethesetoftorus-invariantCartierdivisors.Foraray
τ∈(1)wedefinevτtobethe(unique)primitivelatticepointonτ,i.e.,the
firstlatticepointonτ.Forh∈SF(N,)weset
Dh:=−h(vτ)Vτ∈TNCDiv(X).
(1)∈τForσ∈wegetDh|Uσ=div(e(−lσ)).Themaph→Dhdefinesanisomor-
phismoffreeabeliangroups:
SF(N,)=∼TNCDiv(X).
Note:AWeildivisorDonXisCartieriffD|Uσisprincipalforallσ∈.
Againwehavearightexactsequencewhichisexactiflin(supp())=NR:
0→M→SF(N,)→Pic(X)→0.(1.2)
Pic(X)isalsoafinitelygeneratedabeliangroup.DefinethePicardnumber
ρXofXtobetherankofPic(X).WehaveρX≤rank(Cl(X))=|(1)|−d.
Ifcontainsatleastoned-dimensionalcone,thenPic(X)is(torsion-)free
whichisequivalenttothesplittingofthissequence.IfXisproper,i.e.,is
complete,thenanymaximalconeofisd-dimensional.Inthepropercaseone
canexplicitlycomputethePicardnumberfromthefan(see[Ewa96,V.5.9]and
]).[Eik93Betweentheabovetwosequenceswehaveanaturalcommutativediagram
enducedbytheembeddingTNCDiv(X)⊆TNDiv(X).Exactlyinthecaseofa
nonsingularvarietythesegroupsareequalandthetwodiagramsarenaturally
isomorphic.Ifanymaximalconeinafanisd-dimensional,e.g.,complete,wehave:
Pic(X)=∼H2(X;Z).
impliesThisPic(X)=∼NS(X),(1.3)
whereNS(X)istheN´eron-Severi-Group,i.e.,thegroupofCartierdivisorsmod-
alence.equivalgebraiculoFromnowonletXbeproper.DefineN1(X)asthegroupofCartierdivi-
sorsmodulonumericalequivalence.TwoCartierdivisorsarecallednumerically
equivalent,iftheyhavethesameintersectionnumber(see[Deb01])withevery
curve,i.e.,bothassociatedinvertiblesheafshavethesamedegreeonoftheir
restrictionstoanycurve.Nowwesimplyget:
Pic(X)=∼N1(X).
EspeciallytheabovedefinedPicardn1umberisinthepropercasethesameas
theusualPicardnumberρX=rank(N(X))(see[Deb01,1.3]).
Inthesamewaylet’slookatcurvesforXproper.A1(X)isgeneratedby
theclassesoftheone-dimensionalorbitclosuresVρforρ∈(d−1),suchaρ

1.5.Polytopesandlatticepoints

23

iscalledawall.By[Rei83]or[Mat02]wehaveforeveryirreduciblecurveCin
NowonedefinesN1(X)asthequotientofthegroupof1-cyclesmodulo
XthatCisrationallyequivalenttoρ∈(d−1)aρ[Vρ]foraρ∈N.
nhaveumericalthesameequivinalence,tersectionwherentumwboer1-cycleswithevareerynumericCartierallydivisor.equivalentThisifgivestheya
airingpctionintersenondegeneratePic(X)×N1(X)→Z.
groupHenceofifXinistegernonsingular,relationsthenamongbythe(1.2)wprimitivegetethatgeneratorsN1(X)ofistheraisomorphicysofto.the
LetNE(X)denotetheMoriconeofcurves,i.e.thesetofclassesofeffec-
tivequivealence,1-cyclestheinNab1o(vXe)⊗resultZR.onBecauserationalequivrationalalenceequivofalencecurvesimpliesyieldsnthatumerical
NE(X)=ρ∈(d−1)R≥0[Vρ](1.4)
isaclosedpolyhedralconegeneratedbytheclassesofthewallsofthefan.

1.5Polytopesandlatticepoints
Firsttwoimportantresultsfromconvexgeometry(e.g.,see[Ewa96,Zie95]).
LetS⊆MRbeanysubset.
Theorem1.5.1(Helly’stheorem).Anypointinconv(S)isintheconvex
hullofatmostd+1pointsinS.
Theorem1.5.2(Steinitz’stheorem).Anypointintheinteriorofconv(S)
isintheinterioroftheconvexhullofatmost2dpointsinS.
Nowlet’sdefinethemainobjects:
dronAispolyhecalleddraonpisolytopaefinite.AinpolytoptersectionePofcanalsohalfspacesbecinMRharacterized.Abasoundedthepconvolyhe-ex
hulloffinitelymanypoints.
TheboundaryofPisdenotedby∂P,therelativeinteriorofPbyrelintP.
WhenPisfull-dimensional,itsrelativeinteriorisalsodenotedbyintP.Analo-
gouslytoconesonecandefinefacesandnormalsofapolytope.Ak-dimensional
faceofPiscalledavertexfork=0,anedgefork=1,orafacet,ifitishas
cothesetdimensionV(P),one.theAfacetsfaceofFPofPtheissetFdenoted(P).bAyFp≤olytopP,etheisvalwaerticesysofthePconvformex
hulldiamondofitsprvopertices.ertystatingThebthatoundaryany(isd−coveredb2)-dimensionalyfacets.faceThereofaisdtheso-dimensionalcalled
polytopeiscontainedinexactlytwodifferentfacets.
respPectiviselycalledV(Pa)⊆latticMe.pAolytope,resphomomorphismectivelyr(resp.ationaliso-)pofolytoplatticee,ifpV(olytopP)⊆esisMa,
Qhomomorphism(resp.iso-)oftheassociatedlatticessuchthattheinducedreal
linearhomomorphismmapstheonepolytopeto(resp.onto)theother.
ThereisthesocalledsupportfunctionhPofa(rational)polytopeP⊆MR
ybdefinedhP:NR→R,y→inf{y,x:x∈P}.

24

Chapter1.Fans,polytopesandtoricvarieties

hPisapositivehomogeneous(see[Ewa96,Def.5.5])upperconvexfunction.
ThereexistsacoarsestcompletefanNPinNRcalledthenormalfanofP
suchthathPislinearoneachcone.Thisterminologycomesfromthefollowing
observation:DefineforafaceFofPthenormalconeNP(F)astheunionof
{0}andthesetofinnernormalsofF.IntermsofhPthismeans
NP(F)={y∈NR:hP(y)=y,x∀x∈F},
wherewetaketheclosureofNP(F).Then
NP={NP(F):F≤P}.
NowletPbead-dimensionallatticepolytopewith0∈intP.Apartfrom
thenormalfanthereisamoreobviouspossibilitytodefineacompletefanfrom
P:LetΣP:={pos(F):F≤P}bethefanspannedbyP.
Thesetwoconstructionscanberelatedbythenotionofthedualpolytope
P∗:={y∈NR:y,x≥−1∀y∈P}.
P∗isad-dimensionalrationalpolytopewith0∈intPandverticesinNQ.P∗
doesnothavetobealatticepolytopeagain.
ηF

FWehavetheduality(P∗)∗=P.
Thereisaninclusion-reversingcombinatorialcorrespondencebetweenk-
dimensionalfacesofPandd−1−k-dimensionalfacesofP∗.Forinstance
thedualofasimplicialpolytope,whereanyfacetisasimplex,i.e.,theconvex
hullofdvertices,isasimplepolytope,whereanyvertexisonlycontainedind
facets.Geometrically,ifFisafaceofP,thenitscorrespondingfaceofP∗isgiven
byallinnernormalsofFwhichlieintheaffinehyperplane{y∈NR:y,x=
−1∀y∈F}.ForafacetF≤PweletηF∈NQdenotetheuniqueinnernormal
satisfyingηF,F=−1.Hencewehave
V(P∗)={ηF:F∈F(P)}.
Wehavenowthefollowingimportantrelation
NP=ΣP∗andviceversaNP∗=ΣP.
Thedualoftheproductofdi-dimensionalpolytopesPi⊆Rdiwith0∈intPi
fori=1,2isgivenby
(P1×P2)∗=conv(P1∗×{0},{0}×P2∗)⊆Rd1×Rd2.(1.5)
Itisingeneralanunsolvedproblemtofindequationsorinequalitiesbetween
non-combinatorialinvariantsofPandP∗suchasthenumberoflatticepoints
orthevolume.Forsomepartialrelationsinlowdimensionssee3.7.1.

1.5.Polytopesandlatticepoints

25

Nowlet’sdoashortexcursionintosomemoreorlesswell-knownconvexand
enumerativegeometry(seeforinstance[Sta86]).
SoletfromnowonP⊆MRbealatticepolytopeofdimensionn≤d.
ChooseaZ-basisofMR,soonecanmeasurelengthsandvolumes,and
alsoaregularparametrizationφ:Rn→Pwheretheimageofthecanonical
basise1,...,enisalatticebasisoftheaffinesublatticeaff(P)∩M.Thenthe
Jacobianofφisdenotedbydetaff(P),thisisjustthevolumeofthefundamental
parallopedoftheaffinesublatticeaff(P)∩M.IfFisafacetofPandn=d,
thenonecanalsoprovethatdetaff(F)isthelengthofζF,whereζF∈Nis
definedastheuniqueprimitiveinnernormalofF,i.e.,ζFisthefirstnon-zero
latticepointontheone-dimensionalconeofinnernormalsofF.
NowtherelativevolumeorlatticevolumeofPiswell-definedas
rvol(P):=volRn(φ−1(P))=detvolnaff((PP)),
wherevoln(P)isjustthevolumeofPinthesenseofdifferentialgeometry.
rvol(P)istherebyinvariantunderunimodulartransformationsofMR,soinde-
pendentofthechosenZ-basis.Therelativevolumeisjusttheordinaryone,if
Pisfull-dimensional,i.e.,n=d.
NowrecallthedefinitionoftheanalyticalbarycenterbPofP
Px1dxPxndx
bP:=voln(P),...,voln(P)),
wheretheintegralmustbeunderstoodinthesenseofdifferentialgeometry.
ObviouslyS(P)∈MQisinvariantofthechosenZ-basis.
InthecaseofbP=0thereisthefollowing(coarse)inequalitycalledBlaschke-
Santal´oinequality(see[Lut93,p.165]):
vol(P)vol(P∗)<ωd2,(1.6)
whereωdisthevolumeofthed-dimensionalunitballinRd.
AnimportantdefinitionisthelatticepointenumeratorofP:
eP(k):=|kP∩M|fork∈N.
Thereisthefollowingclassicalresult:
Theorem1.5.3(TheoremofEhrhart).Thereexistsauniquepolynomial
e(X)∈Q[X]calledEhrhartpolynomialofPsuchthate(k)=eP(k)forall
k∈N.Ithasdegreedeg(e)=n=dim(P).Denotecoeffi(e)∈Qforthe
coefficientofe(x)ofdegreei∈N.Thenonehas
coeffn(e)=rvol(P),
coeffn−1(e)=1rvol(F),
2F∈F(P)
coeff0(e)=1.
Andthefollowingreciprocitylawholds:
|relint(kP)∩M|=(−1)de(−k)∀k∈N>0.

26

Chapter1.Fans,polytopesandtoricvarieties

Especiallytherelativevolumeofalatticepolytopeisarationalnumber.By
augmentationthisisalsotrueforpolytopeswithverticesinMQ.
Asanapplicationlet’sprovethetheoremofPick:LetPbeatwo-dimensional
latticepolytopeintheplane.Thenvol2(P)=coeff2(eP)=1/2((eP(1)−1)+
(eP(−1)−1))=1/2(|P∩M|+|intP∩M|−2)=1/2|∂P∩M|+|intP∩M|−1.
RecentlythetheoremofEhrhartwassubstantiallygeneralizedbythefol-
lowingtheoremofBrionandVergne[BV97,Prop.4.1]:
Theorem1.5.4.LetP⊆MRbead-dimensionallatticepolytope.Letφbea
homogeneouspolynomialfunctionofdegreeg.Definefork∈N
i(φ,P)(k):=φ(m)andi(φ,intP)(k):=φ(m).
m∈kP∩Mm∈(kintP)∩M
hasoneThenk→i(φ,P)(k)andk→i(φ,intP)(k)
arepolynomialsofdegree≤d+g.Thereisthereciprocitylaw:
i(φ,intP)(k)=(−1)d+gi(φ,P)(−k)∀k∈N>0.
TorecoverthemainpartsofthetheoremofEhrhartjustchooseφ=1.

1.6BigandnefCartierdivisors
LetX=X(N,)andh∈SF(N,).Wedefinethepolyhedron
Ph:={x∈MR:x,y≥h(y)∀y∈supp()}.
Ifsupp()generatesNRasacone,i.e.,pos(supp())=NR,e.g.,complete,
thenPhisa(possiblyempty)polytopeofdimensionn≤d.Theglobalsections
areDofhH0(X,OX(Dh))=Ce(u).(1.7)
u∈Ph∩M
EspeciallythedimensionoverCoftheglobalsectionsofOX(Dh)isjustthe
numberoflatticepointsinPh.
LetXfromnowonbeproperandhbe(uniquely)givenbylσ∈Monevery
coneσ∈(d).
Itfollowsfrom1.7that,ifPh=∅,then
ePh:k→dimCH0(X,OX(kDh))
isjusttheEhrhardtpolynomialofthen-dimensionalpolytopePhofdegreen
withGenerallyleadingacoefficienCartiertrvoldivisor(PhD).onXiscalledbig,if
liminfdimCH0d(X,kD)>0.
k∞+→kthatthereforewsfolloItDhbig⇐⇒Phhasdimensionn=d.
OnecallsaCartierdivisorDnef,ifD∈NE(X)∨,i.e.,D.C≥0forall
curvesConX.

1.6.BigandnefCartierdivisors

27

Proposition1.6.1.Therearethefollowingequivalences:
1.Dhisbase-point-free,i.e.,OX(Dh)isgeneratedbyitsglobalsections
2.hisupperconvex,i.e.,h(x)+h(y)≤h(x+y)forallx,y∈NR
3.lσ∈Phforallσ∈(d)
4.Ph=conv(lσ:σ∈(d))
5.Dh.Vρ≥0forallρ∈(d−1)
nefisD6.hIfthisholds,thenh(v)=inf{u,v:u∈Ph∩M}forv∈Nandhisthe
supportfunctionofthenon-emptyn-dimensionallatticepolytopePh.
Theseequivalencescanberegardedasastrongversionofthebase-point-
freenesstheorem(see[Deb01,7.32])forpropertoricvarieties.
Inthiscontextweshouldmention,howtocomputetheintersectionnumber
alence:equivfifththeinLetρ∈(d−1).Thereexistσ1,σ2∈(d)withρ=σ1∩σ2.Nowlet
N:=N/<ρ∩N>Z.Onehasthecanonicallyprojectedfan={{0},σ1,σ2}
whereσ1=pos(vσ1)andσ2=pos(vσ2)forprimitivevσ1,vσ2∈N.Wehave
Vρ=X(N,)∼=P1.Forh=h−lσ1∈SF(N,)wehave
Dh.Vρ=deg(OVρ(Dh))=−h(vσ1)−h(vσ2).
Let’slookattheself-intersectionnumberofaCartierdivisorDonX:Dd
isgenerallydefinedasd!-timestheleadingcoefficientofthepolynomial
qD:k→χ(X,OX(kD))
.ddegreeof≤ForDhnef,thereisthefollowingvanishingtheorem(thatisspecialforthe
case):toricHi(X,OX(Dh))=0foralli>0.(1.8)
IfDhisnef,thenthisimplies
qDh=ePh
andthereforeweget(novanishingtheoremisnecessaryforthis,see[Deb01,
1.31])Dhd=d!vold(Ph),
andDhisalsobigiffthisnumberisnotzero.
Goingfurtherintointersectiontheoryandmixedvolumesonecanprovethe
theoremofBernstein(see[Ful93,5.5]).

28

Chapter1.Fans,polytopesandtoricvarieties

1.7AmpleCartierdivisorsandprojectivetoric
arietiesvLetX=X(N,)beapropertoricvarietyandh∈SF(N,)givenbyelements
lσ∈Mforσ∈(d).
LetE={m0,...,ms}⊆Ph∩Mbeanon-emptysetofglobalsectionsof
Dh(undertheusualidentification).Thenwehave:
EgeneratesOX(Dh)ifflσ∈Eforallσ∈(d).
SuchanEexistsiffDhisbase-point-free,i.e.,nef.Inthiscasethereisthe
followingmorphismoftoricvarieties:
ΨE:X→Ps(C),x→[e(m0)(x):∙∙∙:e(ms)(x)](1.9)
Proposition1.7.1.Therearethefollowingequivalences:
1.EgeneratesOX(Dh)andΨEisaclosedimmersion
2.Forallσ∈(d)isΨE|UσaclosedimmersionandΨ−E1({xj=0})=Uσ
formj=lσ
3.SσisgeneratedbyE−lσforallσ∈(d)andhisstrictlyupperconvex,
i.e.,h(x)+h(y)≤h(x+y)forallx,y∈NRwithequalityiffxandyare
containedinacommonconeσ∈(d)
4.DhisampleandSσisgeneratedbyE−lσforallσ∈(d)
ForanysuchEthedegreeofXundertheclosedimmersionΨEisd!vold(Ph).
ToDhgetisaverycriterionampleforiffveryDhisampleness,nefandjustΨPhset∩MEa=Pclosedh∩M:immersion
iffSσisgeneratedbyPh∩M−lσforallσ∈(d)
andhisstrictlyupperconvex.
Howeveramplenessisamorecombinatorialcondition:
Proposition1.7.2.Therearethefollowingequivalences:
1.hisstrictlyupperconvex
2.lσ∈Phforallσ∈(d)andlσ1=lσ2forσ1,σ2∈(d)withσ1=σ2
3.Phhasasverticesexactlythepairwisedifferentelementslσforσ∈(d)
4.Dh.Vρ>0forallρ∈(d−1)
5.Dhisample,i.e.,thereexistsak>0suchthatDkhisveryample
IfDhisample,thenDhisalsonefandbig,especiallyDhisbase-point-freeand
Phisad-dimensionallatticepolytopewithNPh=.
SointhetoriccaseKleiman’scriterionholdsalsoforpropervarieties.One
corollary:aasgetsXisprojectiveiffisspannedbyad-dimensional
latticepolytopeQ⊆NRwith0∈intQ
iffisthenormalfanofad-dimensional
latticepolytopeP⊆MR.

1.7.Projectivetoricvarieties

proDefinitionjectivetoric1.7.3.varietLetyPas⊆

MRbearationalpolytope.eWdefinetheasso29

ciatedXP:=X(N,NP).
AconcretewaytodescribeXPistotakeEasthesetoflatticepointsina
sufficientlylargemultipleofPanddefineXPastheclosureofΨE(C∗)din(1.9).
AnalternativeconstructionisalsopossiblebydescribingXPastheprojective
spectrumofasemigroupalgebra(see[Bat94]).
Ford-dimensionalrationalpolytopesP1,P2equation(1.5)implies

XP1×XP2∼=XP1×P2.
areRemarkisomorphic1.7.4.asAsabstrwasactvobservarietiesedinifand[Con02,onlyProp.ifthey2.1]areprojeasctivetoricvtoricvarieties.arieties

2Chapter

varietiesSingularitiesandtoricFano

ductiontroInInthissectiondesingularizationorresolutionofsingularitiesisdescribedand
ishownotdifferenexplicitlytclassesconoftainedinsingularitiesthestandardcanbederivliteratureedoffromthis.algebraicSincegeometrythissubsucjecth
asHartshorne[Har77],itisatreatedhereinamoredetailedmannerthanabove.
Wereferto[Mat02]and[Deb01].Furthermorewesetupthecorrespondence
cofhapters.toricFanoHerevwearietiesreferandtoF[Dai02anop],olytop[Deb03es,]thisand[Sat02section].isessentialforthenext

2.1Resolutionofsingularitiesanddiscrepancy
LetXbeanormalcomplexvariety.XiscalledgloballyfactorialiffanyWeil
divisorisprincipal(Cl(X)=0).Intheaffinecasethisisjustthefactthatthe
coordinateringisfactorial.RecallthataCartierdivisorisalocallyprincipal
Weildivisor.Thereistheanalogousdefinitionofafactorialvarietyiffany
WeildivisorisCartier.Thisisequivalenttothefactthatanylocalizationisa
factorialring.ThisholdsforXnonsingular.AnalogousonecandefineaWeil
divisorDtobeQ-principal,respectivelyQ-Cartier,ifthereexistsapositive
naturalnumberksuchthatkDisprincipal,respectivelyCartier.IfanyWeil
divisorisQ-principal,respectivelyQ-Cartier,XiscalledgloballyQ-factorial,
respectivelyQ-factorial.FinallyanelementinDiv(X)⊗ZQiscalledQ-divisor.
Remark2.1.1.Itwouldbemoresystematictocallglobally(Q-)factorialva-
rieties(Q-)factorialvarieties,and(Q-)factorialvarietieslocally(Q-)factorial
varieties(thisisdonein[Deb01]).Howeverintoricliteraturethenotionof
Q-factorialvarietiesisalreadyestablished(see[Mat02]or[Cas04]).
LetX,Ybecomplexvarietiesofdimensiond.
GenerallyadesingularizationorresolutionofsingularitiesofXisdefined
asaproperbirationalmorphismf:Y→XwithYnonsingular.Theexistence
ofsucharesolutionispartofthenexttheoremwhichissomewhatstrongerand
notation.moresomeneeds

31

32

Chapter2.SingularitiesandtoricFanovarieties

Firstletf:Y→XbeaproperbirationalmorphismwithXnormal.Define
theexceptionallocusE:=Exc(f)astheclosedsubsetofYwherefisnota
localisomorphism.Thenfissurjective,E=f−1(f(E))andcodimXf(E)≥2.
f(E)isthesetofelementsofXwithpositivedimensionalfibreandX−f(E)
isthedomainoff−1,i.e.,thelargestopensetoverwhichf−1:X−−→Yis
defined.Hereitisimportanttonotethat,ifXisQ-factorialeveryirreducible
componentofExc(f)hascodimension1,i.e.,theexceptionallocusistheunion
ofprimedivisors.Ingeneralthismightnotbetrue.(Fortheseresultssee[Deb01,
1.40]).SecondletYbenonsingularandDaneffectivedivisoronY.Dhassim-
plenormalcrossingsiffeachirreduciblecomponentofDredisnonsingularand
wheneversomeirreduciblecomponentsmeetatapointy,theirlocalequa-
tionsformapartofaregularsystemofparametersinOY,y.Thisjustmeans
thatanysubsetofirreduciblecomponentsofDredintersecttransversally,i.e.,
edimOY,y/(f1,...,fr)=d−rforf1,...,frlocallydefiningequationsofir-
reduciblecomponentsD1,...,DrofDredfor∩i=1,...,rDi=∅.Especiallyany
non-emptyintersectionofirreduciblecomponentsofDredisnonsingular.
Nowthereisthefollowingfundamentalresult(see[Deb01,7.22]):
Theorem2.1.2(Hironaka’stheoremonembeddedresolution).LetX
beacomplexvarietyandZasubschemeofX.Thereexistsanonsingular
complexvarietyYandaprojectivebirationalmorphismf:Y−−→Xsuchthat
Exc(f)∪f−1(Z)isaneffectivedivisorwithsimplenormalcrossings.
Nextlet’sdefineahierarchyofsingularitiesonanormalcomplexvarietyX
:ddimensionofForthiswefirstconstructthecanonicaldivisoronapossiblysingularXby
reductiontothenonsingularcase.LetXreg:=X−Sing(X)bethenonsingular
locus.ThenthereexistsΩdXregthesheafofdifferentialsonXreg,alocallyfree
sheafofrangd.ThereforeωXreg:=∧dΩdXregthecanonicalsheafonXregis
aninvertiblesheaf.ChooseaWeildivisorKXregwithOXreg(KXreg)=ωXreg.
BecauseofcodimXSing(X)≥2thereexistsauniqueWeildivisorKXcalled
canonicaldivisoronXwithKX|Xreg=KXreg.NotethatKXonlyisuniqueup
tolinearequivalence.−KXiscalledanticanonicaldivisor.
Fortheconstructionofthesocalledramificationformulathenextlemma
(see[Deb01,7.11]andproofonpage177)anditscorollariesareessential:
Lemma2.1.3.Letf:Y→XbeaproperbirationalmorphismwithXnormal.
DaCartierdivisoronXandFaneffectiveCartierdivisoronYwhosesupport
iscontainedinE:=Exc(f).Then
H0(X,D)=∼H0(Y,f∗D+F)=∼H0(Y−E,f∗D+F)=∼H0(X−f(E),D).
Corollary2.1.4.Letf,X,Y,Easinthelemma.
1.The(global)sectionsofthestructuresheafsonY,Y−E,X,X−f(E)are
same.the2.Therearenonon-zeroprincipaldivisorsonYwhosesupportiscontained
.Ein

2.1.Resolutionofsingularitiesanddiscrepancy

33

3.LetFbeaCartierdivisoronYwhosesupportiscontainedinE.IfFis
thenctive,effef∗OY(F)=∼OX.
Thereverseimplicationholds,ifYisnonsingular.
4.LetYbenonsingular.IfD,DareCartierdivisorsonXandF,Fare
divisorsonYwhosesupportiscontainedinEwith
f∗D+F≡f∗D+F,
thenD≡DandF=F.
Letf:Y→Xbearesolutionofsingularities,andletKbeanarbitrary
canonicaldivisoronY.Wehave
Y−E=∼X−f(E),OY−E(K|Y−E)=∼OX−f(E)(KX|X−f(E)).
ThereforethereexistsarationalfunctionronYsuchthatforthecanonical
divisorKY:=K+div(r)theWeildivisorKY|Y−EmapstoKX|X−f(E).
BecauseofthesecondpointinthecorollaryKYisuniquelydeterminedbyf
.KandXFinallythereisthefollowingdefinition:
Definition2.1.5.LetKXbeQ-Cartier,i.e.,thereexistsapositiveintegerj
suchthatjKXisaCartierdivisor.Thesmallestsuchjiscalledthe(Gorenstein)
indexjXofX.Thisdefinitionisofcourseindependentofthechosencanonical
.KdivisorXInthiscasethereexistuniquenumbersai∈Zsuchthat
jXKY=f∗(jXKX)+aijXEi,
iwhereEiaretheexceptionaldivisors,i.e.,theirreduciblecomponentsofEof
1.ddimension−AsanequalityofQ-divisorswegetthesocalledramificationformula
KY=f∗KX+aiEi.
iwhereiaiEiiscalledtheramificationdivisor.
IfCX,CYwereanycanonicaldivisorsonX,respectivelyY,with
CY≡f∗CX+aiEi,
ithena=aibythelastpointofthecorollary.Especiallytheaiandthe
ramificationdivisorareindependentofthechosencanonicaldivisoronX.
Sointheramificationformulawecanreplaceequality=bylinearequivalence
≡.IfXandYarebothprojective,thenbyusing[Deb01,7.19]wecanshow
thataCartierdivisoronYwhosesupportiscontainedinEisequalto0iffitis
numericallyequivalentto0.Soitwouldbewell-definedeventoreplacelinear
bynumericalequivalenceintheramificationformula.
Ifforaresolutionftheramificationdivisorvanishes,i.e.,ai=0forall
exceptionaldivisorsEi,e.g.,noexceptionaldivisorexists,thenfiscalleda
crepantresolution.Thisdoesnotimplyotherresolutionsalsotobecrepant.

34Chapter2.SingularitiesandtoricFanovarieties
Nowwearereadytodefineterminalandcanonicalsingularities.
Definition2.1.6.AcomplexvarietyXhasterminal(respectivelycanonical)
singularities,ifXisnormal,thecanonicaldivisorisQ-Cartier,andai>0
(respectivelyai≥0)foralliinagivenresolutionofsingularitiesf.
Thelastconditionisequivalenttotheintrinsicconditionf∗OY(jXKY−
E(1))=∼OX(jXKX)forE(1)theunionoftheexceptionaldivisors(respectively
f∗OY(jXKY)=∼OX(jXKX))ascanbeseenfromthethirdpointofthecorollary
byusingtheprojectionformula.
Becausetwodesingularizationscanalwaysbedominatedbyathird,onecan
showasin[Deb01,7.14]againbyusingabovecorollarythatthisdefinitiondoes
notdependonthechosenresolutionofsingularities.
WhatwehaveseensofarimpliesimmediatelythatifXisQ-factorialand
hasterminalsingularitiesandthereexistsacrepantresolution,thenXmust
nonsingular.ebalreadyNowbyconsideringsocalledboundary-divisorsonecandoverymuchthe
sameintheso-calledlog-situation.Butinsteadofrepeatingherethismore
generalconstructionweusetheunifyingnotionofdiscrepancy:
Definition2.1.7.LetXbenormalwithQ-Cartiercanonicaldivisor.The
discrepancydiscr(X)ofXistheminimumofallaiand1,ifai≥−1forall
someresolutionofsingularitiesfsuchthatE(1)=iEiisaneffectivedivisor
i,and−∞otherwise,wheretheaiaredefinedintheramificationformulafor
crossings.normalsimplewithSucharesolutionexistsbyHironaka’stheoremandthisnotionisindependent
desingularization.hosenctheofspXectivelyhasdiscrterminal≥0).(respEvenectivtuallyelywecanonical)alsodefine:singularitiesiffdiscr(X)>0(re-
ifdiscrDefinition(X)>−2.1.8.1(respXhasectivloelygdiscrterminal≥−(resp1).ectivelylogcanonical)singularities,
Someremarksconcerningthenotion’logterminal’shouldbemade.There
aremanytechnicaldefinitionsaround.Theyallinvolvealog-pair(X,D)where
[0D,is1].aTsoogetcalledthebcaseoundaryabovedivisor,justseti.e.,DaQ=0.-divisorInwheregeneralallonecodefinesefficientsthearedis-in
crepancyslightlydifferentinanintrinsicmanner(see[Mat02,4.4.1])togetthe
strongestnotioncalled’purelylogterminal’asin[Mat02,4.4.2].Theabove
notionisactuallytheweakernotion’logterminal’whichisingeneralnotinde-
phandendeniftallofcotheefficiengiventsoftheresolution,boundaryforthisdivisorseeare[Mat02<,1(asin4.3.2,4.3.3].thecaseOnhere)theotherthese
tw4.4.3]oanddefinitions[Deb01,coincide7.24]),withaonenotioncalledwhic’khaiswamataagainlogindepterminal’endentof(seethe[Mat02given,
resolution,asdescribedin[Deb01,7.25].
Finallyoneshouldremarkthatthesedefinitionsgivenaboveareindeedlocal.
singularIndimension([Mat02,two4.6.5]).aInnormaldimensionsurfacehasthreeorterminalhigherasingularitiesnormaliffvitarietisynon-with
terminalsingularitiessatisfiescodimSing(X)≥3([Mat02,4.6.6]).

2.2.Singularitiesontoricvarieties35
2.2Singularitiesontoricvarieties
Inthissectionitwillbeshortlydescribed,howtocharacterizeabovedefined
singularitiesontoricvarietiesusingtheresultsofthepreviouschapter(see
]).[Dai02and][Bor00SoletX=X(N,).WedefineGσ:={vτ:τ∈σ(1)}forσ∈.
ThestrongestconditionisactuallyXbeinggloballyfactorial.Thisisequiv-
alenttothefactthat{vτ:τ∈(1)}ispartofaZ-basisofN.Thisholdsiff
VτisInatheprincipaltoriccasedivisorthereforarealltheτ∈follo(1)wing.specialequivalences:
Proposition2.2.1.Thefollowingconditionsareequivalent:
nonsingularisX1.2.GσispartofaZ-basisofNforallσ∈
3.Uσ=∼Cdimσ×(C∗)d−dimσforallσ∈
4.Uσisgloballyfactorialforallσ∈
5.VτisCartierforallτ∈(1)
factorialisX6.Inlowdimensionsonecantrytofindclassificationresultsofisomorphism
classesofpropernonsingulartoricvarietiesuptosomePicardnumberρX=
|(1)|−d.Inthetwo-dimensionalcasethesetofisomorphismclassesarein
bijectionwithaspecialsetofso-calledweightedcirculargraphs,see[Oda88,
1.29].Usingtheseresultsthree-dimensionalpropernonsingulartoricvarieties
withPicardnumberfiveorlesswhichareminimalinthesenseofequivariant
blow-upswerecompletelydescribedin[Oda88,1.34].
Analogouslylet’sexaminethe’Q-case’:
XisgloballyQ-factorialiff{vτ:τ∈(1)}islinearlyindependentiffVτis
Q-principalforallτ∈(1).
Proposition2.2.2.Thefollowingconditionsareequivalent:
1.issimplicial,i.e.,|σ(1)|=dim(σ)forallσ∈
2.Gσislinearlyindependentforallσ∈
3.UσisgloballyQ-factorialforallσ∈
4.VτisQ-Cartierforallτ∈(1)
-factorialQisX5.Ournextgoalistodescribetheresolutionofsingularitiesinthetoriccase.
Forthisoneneedsatoricdescriptionofthecanonicaldivisor:
KX:=−Vτ
(1)∈τisacanonicaldivisor(see[Mat02,14.3.1]).Hereoneshouldnotethatanytoric
varietyisCohen-Macaulay,andthiscanonicaldivisordefinesthedualizingsheaf
ofX.FurthermoreXalwayshasrationalsingularities.Forthesenotionssee
[Ful93,pp.30,76,89]and[Dai02].

36

Chapter2.SingularitiesandtoricFanovarieties

Remark2.2.3.KXisQ-Cartieriffforallσ∈thesetGσiscontainedin
anaffinehyperplane,orequivalently,iffforallσ∈thereexistsanuσ∈MQ
suchthatuσ,vτ=1forallτ∈σ(1).
Asdescribedin[Ewa96,VI.8.5]onecanrefinebyfinitelymanysocalled
stellarsubdivisionsthefanintoanonsingularfanwith(1)⊆(1)so
thatf:X:=X(N,)→Xisatoricmorphismandmoreoveraresolution
ofsingularities.Thenwegetthat
E(1)=Vτ
τ∈(1)−(1)
isaneffectivedivisoronXwithsimplenormalcrossings.
ItshouldbenotedthatthereisatoricChow’slemma,i.e.,ifXisproper,
onecanchoosefsuchthatYisprojective.
LetKXbeQ-Cartier.Thentheramificationformulalookslikethis:
−jXVτ=jXf∗KX+jX(−1+uσ,vτ)Vτ,
τ∈(1)τ∈(1)−(1),τ⊂σ∈
becausejXKX|Uσ=div(jXe(uσ))forallσ∈anduσasin2.2.3.Therefore
discr(X)=−1+σ∈maximal,0min=v∈σ∩N−Gσuσ,v
isarationalnumberin]−1,1](seealso[Deb03,Prop.12]).
Proposition2.2.4.LetKXbeQ-Cartier(anduσdefinedasin2.2.3).Then
Xhaslogterminalsingularities.
Xhasterminalsingularitiesiff{x∈σ:uσ,x≤1}∩N={0}∪Gσ
forallσ∈maximal.
Xhascanonicalsingularitiesiff{x∈σ:uσ,x<1}∩N={0}
forallσ∈maximal.
TheresolutionfaboveiscrepantiffforallexceptionaldivisorsVτwehave
vτ∈aff(Gσ)forτ⊂relint(σ)andsomeσ∈.
Ifallmaximalconesofared-dimensional,thentheGorensteinindexjX
istheleastcommonmultipleofall(positive)denominatorsofthecoefficientsof
thevectors{uσ}σ∈(d).
TheexistenceofacrepantresolutionimpliesjX=1.

2.3ToricFanovarieties
HerewelayoutthebasicnotionsoftoricFanovarieties.Forasurveysee
].[Sat02and][Deb03Definition2.3.1.AcomplexvarietyXiscalledFanovariety(respectively
weakFanovariety),ifXisprojective,normalandtheanticanonicaldivisor
−KXisanample(respectivelynefandbig)Q-Cartierdivisor.
BecauseFanovarietiesareinsomesenseopposedtovarietiesofgeneraltype,
i.e.,wherethecanonicaldivisorKXisample,theyarequiterareandobjects
ofmajorstudies(see[Deb01,Deb03,PSG99]).

2.3.ToricFanovarieties37
Insomesources(e.g.,[Deb03])aFanovarietyisassumedtobelogterminal.
Sinceweareonlyconcernedwiththetoriccase,thisassumptionisredundant
2.2.4.Prop.ybNowtothetoriccase:LetbeacompletefaninNR,andX:=X(N,)
theassociatedcompletenormaltoricvariety.
WesetG:={vτ:τ∈(1)}andGσ:=G∩σforσ∈asinthe
section.previousNowwedefinethelatticepolytopeQ∗:=conv(G)⊆NRwith0∈intQ,
andtherationaldualpolytopeP:=Q⊆MR.
Fromtheresultsofthelastchapterwecanimmediatelyderivethefollowing
ositions:propProposition2.3.2.Thefollowingconditionsareequivalent:
1.XisatoricweakFanovariety
2.τ∈(1)VτisanefQ-Cartierdivisor
3.Thereexistsk∈N>0andanupperconvexh∈SF(N,)suchthath(vτ)=
−kforallτ∈(1)
4.conv(Gσ)generatesanaffinehyperplaneforallσ∈(d)andanyfacet
ofQconsistsofanunionofconv(Gσ)forsomeσ∈(d)
Ifthisholds,thentheminimalkisjusttheGorensteinindexjX,andwhenh
isgivenby{lσ},then
jXP=conv(lσ:σ∈(d))=Ph,
withV(P)={lσ/jX:σ∈(d)}.EspeciallyjXisalsotheminimalksuch
thatkPisalatticepolytope.
Proposition2.3.3.Thefollowingconditionsareequivalent:
1.XisatoricFanovariety
2.τ∈(1)VτisanampleQ-Cartierdivisor
3.Thereexistsk∈N>0andastrictlyupperconvexh∈SF(N,)suchthat
h(vτ)=−kforallτ∈(1)
4.Forallσ∈(d)thepolytopeFσ:=conv(Gσ)isafacetofQ
5.=ΣQ
6.=NP
InthiscaseV(Q)=GandV(P)={lσ/jX:σ∈(d)}with|V(P)|=
|(d)|.
Example2.3.4.Inthefollowingfigurewehavefansthatdefineinthisorder:
notaweakFanovariety;aFanovariety;justaweakFanovariety;aFano
variety.HerebytheblackpolygonisQandtheblackdotsformthesetG:

38

Chapter2.SingularitiesandtoricFanovarieties

Thefollowing(hierarchicallydescending)definitionsarenowconvenient:
Definition2.3.5.LetQ⊆NRbead-dimensionallatticepolytopecontaining
theorigininitsinterior.
∙QiscalledaFanopolytope,iftheverticesareprimitivelatticepoints.
∙QiscalledacanonicalFanopolytope,ifintQ∩N={0}.
∙QiscalledaterminalFanopolytope,ifQ∩N={0}∪V(Q).
∙QiscalledasmoothFanopolytope,iftheverticesofanyfacetofQform
aZ-basisofthelatticeN.

Remark2.3.6.Beware:Inmostpapers(see[Bat99,Sat00])aFanopolytope
isassumedtobealreadyasmoothFanopolytope.Thismoresystematicnota-
tionthatwillbeusedthroughoutthisthesiswaspartlyintroducedin[Deb03].
Moreoverinpolytopeliteraturesometimesapolytopeiscalled’smooth’,ifit
hasaspecialsymmetrypropertyorevenifthedualisasmoothFanopolytope.
Soonealwayshastocheckthedefinitionscarefully!
Wehavethefollowingcorrespondencetheorem(see1.7.4and2.2.4):
Proposition2.3.7.Thereisacorrespondencebetweenisomorphismclassesof
FanopolytopesandisomorphismclassesoftoricFanovarieties.
TherebycanonicalFanopolytopescorrespondtotoricFanovarietieswith
canonicalsingularities,i.e.,discrX≥0;
terminalFanopolytopescorrespondtotoricFanovarietieswithterminal
singularities,i.e.,discrX>0;
smoothFanopolytopescorrespondtononsingulartoricFanovarieties.
Example2.3.8.Thereareinfinitelymanynon-isomorphicd-dimensionalFano
polytopesford≥2asthefollowingexamplesford=2show.Herebythe
polygonisterminalandsmoothfork=1,canonicalbutnotterminalfork=2,
andnotcanonicalfork≥3:

(k,1)

2.3.ToricFanovarieties

39

Inanydimensionthereisuptoisomorphismalwaysonlyafinitenumber
ofcanonicalFanopolytopes,asfollowsfromthefollowingimportantfiniteness
theorem(see[Bor00]forasurvey,thisresultcanbededucedfrom3.7.14):
Theorem2.3.9.For>0thereexistonlyfinitelymanyisomorphismclasses
ofd-dimensionaltoricFanovarietieswithdiscrepancygreaterthan−1+.
Fortheweakcasewedefine:
Definition2.3.10.LetQ⊆MRbeaFanopolytopespanning.Thena
faniscalledacrepantrefinementof,ifisarefinementofinthe
usualsenseandadditionallyforanyτ∈(1)thereexistsaσ∈such
thatvτ⊆conv(Gσ).Whenthetoricvarietyassociatedtothefanisagain
projective,thefaniscalledacoherentcrepantrefinement.
Usingtheramificationformula(asonpage36)weseethatsuchcrepant
refinementscorrespondtoequivariantproperbirationalmorphisms
f:X=X(M,)→X=X(M,)withKX=f∗KX.
Proposition2.3.11.ToricweakFanovarietiescorresponduniquelyuptoiso-
morphismtocoherentcrepantrefinementsoffansspannedbyFanopolytopes.
Forthenextresult(seeProp.2.2.2forthefirstand[Bat94,Thm.2.2.24]
forthesecondpart)recallthatapolytopeissimplicial,ifanyfacetisasimplex:
Proposition2.3.12.Q-factorialtoricFanovarietiescorresponduniquelyup
toisomorphismtosimplicialFanopolytopes.
Thereexistsacoherentcrepantrefinementbystellarsubdivisionsthatre-
solvesaweaktoricFanovarietyXwithcanonicalsingularitiestoaQ-factorial
weaktoricFanovarietyXwithterminalsingularities.
SuchamorphismX→XiscalledaMPCP-desingularizationin[Bat94].
Xissaidtoadmitacoherentcrepantresolution,ifsuchanXcanbechosento
nonsingular.ebFinallywecometotheessentialnotion:
Definition2.3.13.AcomplexvarietyXiscalledGorenstein,iffjX=1,i.e.,
KXisaCartierdivisor.
ThereareothercharacterizationsoftheGorensteinpropertyintermsofthe
localrings[Eis94,Ch.21]andtheaffinesemigrouprings[Oda88,p.126,Remark
(i)].Forourpurposesthisdefinitionisthemostsuitableone,wereferto[Bat94]
A].App.,[CK99andNowlet’slookalittlebitmorecloselyatthelocalsituation:
Proposition2.3.14.Thefollowingconditionsareequivalentforσ∈(d):
GorensteinisU1.σ2.τ∈σ(1)Vτisaprincipaldivisor
3.Thereexistslσ∈Msuchthatlσ,vτ=−1forallτ∈σ(1)
4.TheuniqueinnernormalηFofthefacetF:=conv(Gσ)ofconv(0,Gσ)is
ointpelattica

40

Chapter2.SingularitiesandtoricFanovarieties

5.ThereexistsaZ-basise1,...,edofNsuchthatGσ⊆{x∈NR:xd=−1}
6.Anyv∈GσcanbeextendedtoaZ-basise1,...,ed−1,vsuchthatGσ⊆
{x∈NR:xd=1}
EspeciallyN∩{x∈σ:lσ,x>−1}={0}.
Proof.3.⇒4.,3.⇔6.stemsfromthesplittingoftheshortexactsequence
0→{x∈N:lσ,x=0}→Nlσ→,∙Z→0.
immediate.areimplicationsotherThe

Intheglobalcasewehavethefollowingequivalences:
Proposition2.3.15.Thefollowingconditionsareequivalent:
1.XisaGorensteintoricFanovariety
2.τ∈(1)VτisanampleCartierdivisor
3.Thereexistsastrictlyupperconvexh∈SF(N,)suchthath(vτ)=−1
forallτ∈(1)
4.QspansthefanandPisalatticepolytope
ThenV(P)={lσ:σ∈(d)}with|V(P)|=|(d)|.
Wehaveby(1.6)fortheanticanonicaldegreeofX:
deg(X):=(−KX)d=d!vol(P).

Asanimportantcorollaryfrom2.3.14and2.3.9weget:
Corollary2.3.16.Gorensteintoric(weak)Fanovarietieshavecanonicalsin-
classesgularities.ofdInfixe-dimensionalddimensionGordensteintheretoricisFonlyanoafinitevarieties.numberofisomorphism

3Chapter

esolytoppeReflexiv

ductiontroInvInthisarieties.cWhaptereinvtheestigatemainobthisjectsclassofofstudyvarietiesareinbtroycomduced:binatorialGorensteinmethotoricdsFusingano
thenotionofareflexivepolytopewhichappearedinconnectiontomirrorsym-
metry(see[Bat94]and[CK99]).Areflexivepolytopeisjustalatticepolytope,
thatdualpconolytoptainseisasalsotheaonlylatticelatticepolytoppoine.ttheThereoriginisupintoitsinisomorphismterior,suchonlyathatfinitethe
numbThisercofhapterreflexivconeptainsolytopesingeneralizationsfixedofdimension.toolsandresultspreviouslyknown
asonlywellforasanonsingulardetailedtorictreatmenFanotofvanarietiesapproac(duehbtoyBatCasagrandeyrevandandConrads.Debarre),As
applicationsweobtainnewclassificationresultsandboundsofinvariants,and
weformulateconjecturesconcerningcombinatorialandgeometricalpropertiesof
theresultsarbitrary-dimensionalachievedherereflexivwillebepolytopapplied.es.Intheremainingpartsofthisthesis
pInolytopesection(see1wProp.egive3.1.4)theanddefinitiondescribeanditssevbasiceralcproperties.haracterizationsofareflexive
eralizedInsectionsthatw2ereand3alreadytwoelemenpreviouslytaryusedtectohnicaltosuccessfullyolsareininvvestigateestigatedandandclassifygen-
nonsingulartoricFanovarieties[Bat99,Sat00,Cas03b].
Insection2weinvestigatetheprojectionofareflexivepolytopealongalat-
ticepointontheboundary(seeProp.3.2.2).Therebywecanrelateproperties
ofrietay(e.g.,GorensteinCor.toric3.2.5).FanoAsvanarietytoapplicationthatofwealogivweanew,er-dimensionalpurelytoriccomFanobinatorialva-
proof(Cor.3.2.8)ofaresultduetoBatyrev[Bat99,Prop.2.4.4]statingthat
Ftheanoanvarietyticanonicalisalwaclassysnofaumericallytorus-inveffectivariante,andprimewedivisorgeneralizeofathisnonsingularresulttotoricthe
caseoflocallycompleteintersections(Prop.3.2.9).
Insection3weconsiderpairsoflatticepointsontheboundaryofareflexive
polytopeandshowthatinthiscasethereexistsageneralizationofthenotion
ofaprimitiverelationasintroducedbyBatyrevin[Bat91](Prop.3.3.1).As
anreflexiveapplicationpolytopwe,einproveparticularthatontherethearediameterconstrainoftstheontheedge-graphcomofabinatoricssimplicialofa

41

42

Chapter3.Reflexivepolytopes

reflexivepolytope(Cor.3.3.2).Therebywegetthatcertaincombinatorialtypes
ofpolytopescannotberealizedasreflexivepolytopes(Cor.3.3.3,Cor.3.3.4).
Insection4wegiveashortreviewofclassificationresultsofreflexivepoly-
topesinlowdimensions.Inparticularwegiveaconciseproofoftheclassifica-
tionofreflexivepolygons(Prop.3.4.1).Furthermorewedescribethealgorithm
thatwasusedbyKreuzerandSkarkeforthecomputerclassificationofreflexive
polytopesindimensionthreeandfour(see[KS97,KS98,KS00]).
Insection5weareconcernedwiththemaximalnumberofverticesofa
reflexivepolytope,respectivelyasimplicialreflexivepolytope.Inthegeneral
caseweformulateaconjecture(Conj.3.5.2),thatwillbeproveninthecaseofa
centrallysymmetricsimplereflexivepolytopeinthelastchapter(Thm.6.2.2).
Herewegivecoarseboundsthatdependonthemaximalnumberofvertices
ofProp.afacet3.5.5).byWealsogeneralizingextendresultstheofVconjectureoskresenskij,ontheKlymaximalachkonumandberofDebarrevertices(see
ofasmoothFanopolytopetothecaseofasimplicialreflexivepolytope(Conj.
3.5.7).Oneofthemainresultsofthissection,thatisevennewinthesmooth
case,istheverificationofthisconjectureundertheassumptionofanadditional
symmetryofthepolytope(Thm.3.5.11).Forthiswegeneralizearesultdueto
Casagrande[Cas03a,Thm.2.4]tothecaseofaQ-factorialGorensteintoricFano
variety(Cor.3.5.17)statingthatthePicardnumberofanonsingulartoricFano
varietyexceedsthePicardnumberofatorus-invariantprimedivisoratmostby
three.Theseresultswerepublishedin[Nil04a]asapreprintinMay2004.In
November2004Casagrandepublishedin[Cas04]theproofoftheconjecturein
thecaseofasmoothFanopolytope.Theauthorpointedouttoherapossible
simplificationinherargumentandinDecember2004Casagrandepublisheda
completeproofofConj.3.5.7partlyrelyingonresultsofthissection,especially
onThm.3.5.11.Fromherproofwecanalsoderiveaboundonthenumber
ofverticesofageneralreflexivepolytope(Cor.3.5.13),therebyimprovingon
3.5.5.Prop.ofBatInyrevsectionand6weConradsdealhowithwtoreflexivdetermineesimplices.theseWsimplicesedescribbyesothecalledwapproaceighht
systems.Thenweprovesharpupperboundsonthetotalweightoftheseweight
systems(Thm.3.6.22).Theproofreliesonsomenumber-theoreticinequalities
fractions.unitofsumsconcerningInsection7weareconcernedwiththenumberoflatticepointsinareflexive
polytope.ReflexivepolytopescanbecharacterizedbyitsEhrhartpolynomial
thatcountslatticepointsinmultiplesofthepolytope(Prop.3.7.2).Themain
theoremofthissectionisasharpupperboundonthevolumeofareflexive
simplex(Thm.3.7.13),thecorrespondingalgebraic-geometricresultyieldsa
generalconjectureonthemaximalanticanonicaldegreeofaGorensteinFano
varietywithcanonicalsingularities.Wealsoobtainaboundonthenumber
oflatticepointsofareflexivesimplex,andfurthermoreweprovethatinany
dimensionthereisauniquereflexivesimplexwiththemaximalnumberoflattice
pointsonanedge(Thm.3.7.19),thiswasobservedinlowdimensionsin[HM04].
Attheendofthissectionwecountlatticepointsmoduloanaturalnumber,and
showhowthisisusefulinthecaseofterminalreflexivepolytopes(Cor.3.7.23).
Summaryofmostimportantnewresultsofthischapter:
∙Propertiesofprojectingreflexivepolytopes(Prop.3.2.2,p.48)

3.1.ertiespropBasic

43

∙Theanticanonicalclassofatorus-invariantprimedivisorofaQ-factorial
GorensteintoricFanovarietywithterminalsingularitiesisanefQ-Cartier
50).p.3.2.7,(Cor.divisor∙Theanticanonicalclassofatorus-invariantprimedivisorofatoricFano
varietywithterminalsingularitiesandlocallycompleteintersectionsisa
nefCartierdivisor(Prop.3.2.9,p.51).
∙Propertiesofpairsoflatticepointsontheboundary(Prop.3.3.1,p.52)
∙Thediameteroftheedge-graphofasimplicialreflexivepolytopesisat
mostthree(Cor.3.3.2,p.53)
∙Proofofsharpupperboundonthenumberofverticesofasimplicial
reflexivepolytopeinthecaseofacentrallysymmetricpairofverticesof
thedualpolytope(Thm.3.5.11,p.61)
∙ThePicardnumberofaQ-factorialGorensteintoricFanovarietyexceeds
thePicardnumberofatorus-invariantprimedivisoratmostbythree
66)p.3.5.17,(Cor.∙LetVbetheanticanonicaldegreeofaGorensteintoricFanovarietyXof
classnumberone,andV∗theanticanonicaldegreeofthe”dual”variety
X∗.ThenweprovesharpboundsonV(Thm.3.7.7,p.83),onV∙V∗
(Thm.3.7.11,p.84),andontheanticanonicaldegreeofatorus-invariant
curveonX(Thm.3.7.9,p.84).

ertiespropBasic3.1Reflexivepolytopesnaturallyenterthepictureasthecombinatoricalcounter-
partsofGorensteintoricFanovarieties.
Definition3.1.1.AFanopolytopewhosedualisalatticepolytopeiscalled
reflexivepolytope.
ByProposition2.3.15thenextresultisstraightforward:
Theorem3.1.2.GorensteintoricFanovarietiescorresponduniquelyupto
isomorphismtoreflexivepolytopes.
GorensteintoricweakFanovarietiescorresponduniquelyuptoisomorphism
tocoherentcrepantrefinementsoffansspannedbyreflexivepolytopes.
From2.3.16weget:
Theorem3.1.3.Areflexivepolytopeiscanonical.Thereareonlyfinitelymany
equivalenceclassesofreflexivepolytopesinanygivendimension.
Thisfinitenessresultisoneofthemainreasonswhythestudyofreflexive
polytopesisontheonehandverymotivating,becauseclassificationresults
shouldbeatleastinverylowdimensionsachievable,andontheotherhandso
difficultbecauseofthe’sporadic’natureofreflexivepolytopes.
Itwouldbeinterestingtofindadirectproofofthisresultthatonlyusesthe
definitionofareflexivepolytope.
HereisanexampletoillustratethedifferenceofcanonicalFanopolytopes
andreflexivepolytopes,itistakenfrom[KS96](see3.2.6foranotherexample):

44

(1,0,0)

Chapter3.Reflexivepolytopes
(1,1,2)

(0,1,0)

(−1,−1,−2)Theouterpolytopeisacentrallysymmetricreflexivepolytope,howeverthe
interioroneisacentrallysymmetricterminalpolytopethatisnotreflexive,in
factallfacetshavelatticedistancetwofromtheorigin,i.e.,theuniqueprimitive
outernormalhasvalue2onthedefiningfacet.
ItisobviousthatlatticepointsontheboundaryofacanonicalFanopolytope
areprimitive,forareflexivepolytopehoweverevenmoreistrue.Thenextresult
summarizesthemostimportantequivalencesofreflexivity(8.and12.seemto
benotyetwrittendownsomewhereelse):
Proposition3.1.4.LetP⊆MRbead-dimensionallatticepolytopewith0∈
.PtineflexiverisP1.2.P∗isalatticepolytope
3.P∗isreflexive
4.X(M,ΣP)isa(Gorensteintoric)Fanovariety
5.X(N,NP)isa(Gorensteintoric)Fanovariety
6.Allfacetshaveintegrallatticedistanceonefromtheorigin(i.e.,forany
F∈F(P)thereexistsνF∈NsuchthatνF,m=1forallm∈F)
7.Therearenolatticepointslyingbetweentheaffinehyperplanespannedby
afacetanditsparallelthroughtheorigin
8.Anylatticepointonanaffinehyperplanespannedbyafacetisprimitive
9.ηF∈NforanyF∈F(P)
10.ζF=ηFforanyF∈F(P)
11.ForanyF∈F(P)andm∈F∩MthereisaZ-basise1,...,ed⊥ofMsuch
thated=mandF⊆{x∈MR:xd=1}(inparticularM∩ηFhasasa
Z-basise1,...,ed−1,andηF=−ed∗inthedualZ-basise1∗,...,ed∗ofN)
12.ForanyF∈F(P)thereisaZ-basise1,...,edofMsuchthatηF=1
andm=1forsomem∈F∩M(thenormtakenwithrespecttothis
basis,respectivelyitsdual)

ertiespropBasic3.1.

45

Proof.8.⇒6.:LetνF:=−ζF∈Nbetheuniqueprimitiveouternormal,
sothereexistsaprimitivelatticepointm∈MsuchthatνF,m=1.Let
c:=νF,F∈N>0.Thencmisalatticepointinaff(F),soprimitive,hence
1.=cTheotherimplicationsandequivalencesareobviousorfollowfromthere-
sultsofthelastsectioninthepreviouschapter,inparticularfromPropositions
2.3.15.and2.3.14

Forimportantalgebraic-geometriccharacterizationsofreflexivepolytopes
(genericanticanonicalhypersurfacesareCalabi-Yau)see[Bat94,Thm.4.1.9]or
[HM04,2.1],foritsconsequencesinmirrorsymmetrysee[Bat94]or[CK99],for
othercombinatorialequivalencesseesubsection3.7.1.
Corollary3.1.5.P→XP:=X(N,NP)inducesacorrespondenceofisomor-
phismclassesofreflexivepolytopesandGorensteintoricFanovarieties.
Inparticularthereisacompletedualityofreflexivepolytopes.Thisimplies
anatural(mirror)symmetryofisomorphismclassesofGorensteintoricFano
varieties.InthefollowingexampletheleftfandefinesP2,therightaquotient
P2/Z2:

IfPisareflexivepolytope,thenXPisanonsingulartoricFanovarietyif
andonlyifP∗isasmoothFanopolytope.
Itisalsoimportanttonote(e.g.,see(1.5))thatproductsofreflexivepoly-
topesareagainreflexive.
Inlowdimensionsreflexivepolytopeshaveaveryimportantandpeculiar
property(see[Bat94]):
Proposition3.1.6.Ford≤3anyd-dimensionalGorensteintoricFanovariety
admitsacoherentcrepantresolution(see2.3.12).
Forafour-dimensionalcounterexamplesee3.2.6.
Werecallaconvenientdefinitionfortheproof:
Definition3.1.7.AlatticepolytopeP⊆MRiscalledempty,ifP∩M=V(P).
Now3.1.6followsfrom2.3.12andthenextwell-knownlemma:
Lemma3.1.8.LetPbead-dimensionallatticepolytopewith0∈intP.
1.Letd=2.Latticepointsx,yformaZ-basisifandonlyifconv(0,x,y)
isanemptytwo-dimensionalpolytope.IfPisacanonicalFanopolytope,
thisholds,ifx,yarelatticepointsontheboundarythatarenotcontained
inacommonfacetofPandx+y=0.
Inparticularanytwo-dimensionalterminalFanopolytopeisasmooth
Fanopolytope,andanytwo-dimensionalcanonicalFanopolytopeisare-
e.olytoppflexive

46

Chapter3.Reflexivepolytopes

2.Letd=3andPbereflexive.Threelatticepointsx,y,zinacommon
facetofPformaZ-basisifandonlyifconv(x,y,z)isempty.
isInapsmoarticularothFanoanypthreolytope.e-dimensionalsimplicialterminalreflexivepolytope

Itiswell-knownthatampleCartierdivisorsonacompletenonsingulartoric
v[EW91ariety]areor[Paalreadyy04],vforeryaamplerelated(seeconjecture[Oda88,seeCor.2.15]).[Kan98].ForHereawegeneralprovbeaoundresultsee
thatseemstobefolklore:
Proposition3.1.9.LetXbead-dimensionalGorensteintoricFanovariety
.3dwith≤Thentheampleanticanonicaldivisor−KXisveryample.
∗Proof.LetnLet∈Ppos⊆(PM∗R−theu)∩Ncorresp.Bytheondingresultsreflexivinepsectionolytop1.7e,uwe∈haV(vPeto)showarbitrarythat.
nisalinearcombinationwithpositiveintegers∗ofelementsin(P∗∩N)−u.
convIt(u,isx1,easyx2)toisseeanthatemptytherelatticeexistpFolytop∈Fe(Pand)nand∈px1,os(0x2−∈u,Fx∩1−Nu,sucx2h−thatu).
By3.1.8(2)wegetthat{u,x1,x2}isaZ-basisofN.Inparticularnisalinear
combinationof{−u,x1−u,x2−u}withpositiveintegers.

3.2Projectingalonglatticepointsonthebound-
aryThroughoutthesectionletPbead-dimensionalreflexivepolytopeinMR.
TheprojectionmapalongavertexofPisanessentialtoolininvestigating
resptoricFondinganovlowarieties,er-dimensionalsinceonevariethopyesto(seeget[Bat99some,Cas03ainformation]).Inthefromcasetheofcor-a
reflexivepolytopeitisalsousefultoconsiderprojectingalonglatticepointson
theboundaryofPthatarenotnecessarilyvertices.
Thefollowingdefinitionswillbeusedthroughoutthiswork:
Definition3.2.1.Letx,y∈∂Pwithx=y.
∙[x,y]:=conv(x,y),]x,y]:=[x,y]\{x},]x,y[:=[x,y]\{x,y}.
∙x∼y,if[x,y]iscontainedinafaceofP,i.e.,xandyarecontainedina
.Poffacetcommon∙Thestarsetofxistheset
st(x):={y∈∂P:x∼y}={F∈F(P):x∈F}.
∙Thelinkofxistheset
∂x:=∂st(x)={G≤P:G⊆st(x),x∈G}.

3.2.Projectingalonglatticepointsontheboundary47
∙yx(yissaidtobeawayfromx),ifyisnotintherelativeinteriorof
Hence).xst(∂x={y∈st(x):yx}.
yisawayfromxiffthereexistsafacetthatcontainsybutnotx,e.g.,if
yisavertexorx∼y.Thereisalsoalocalcriterion:
yx⇐⇒x+λ(y−x)∈P∀λ>1.
Thenextpropositionshowstheimportantpropertiesoftheprojectionofa
reflexivepolytopealongsomelatticepointontheboundary.
Proposition3.2.2.LetP⊆MRbeareflexivepolytope.Letv∈∂P∩M.
WedefinethequotientlatticeMv:=M/Zvandthecanonicalprojection
valongmapπv:MR→(Mv)R=MR/Rv.
ThenPv:=πv(P)isalatticepolytopein(Mv)RwithV(Pv)⊆πv(V(P))that
containstheorigininitsinterior.
1.LetUbethesetofelementsx∈Psuchthatx+λv∈Pforallλ>0.
TherestrictionofπvtoUinducesabijection
.PUv→Wedenotetheinversemapbyιv.
ForS:=st(v)wehave
S=UthusandPv=conv(πv(V(P)∩∂v)).
2.TheprojectionmapinducesabijectionS∩M→Pv∩Mv.
3.Theprojectionmapinducesabijection∂v→∂Pv.∂Pviscoveredbythe
projectionofall(d−2)-dimensionalfacesCsuchthatC∈F(F)forsome
facetFofPwithv∈Fandv∈C;πv(C)iscontainedinafacetofPv.
4.ιv(V(Pv))⊆V(P)∩∂v.
Letz∈V(P)∩∂v.Thenπv(z)∈∂Pv.If]v,z[iscontainedintherelative
interiorofafacetofP,thenπv(z)∈V(Pv).
5.LetF∈F(P)withv∈F.Then
pos(πv(F))∩Pv=πv(F).
6.Theimageπv(F)ofafacetFparalleltov,i.e.,ηF,v=0,isafacetof
Pv.Itisπv−1(πv(F))∩P=F.Thereareatleast|V(πv(F))|-verticesof
FinS.AnypointinF∩Siscontainedinafacetthatcontainsvand
intersectsFina(d−2)-dimensionalface.
ThepreimageΓ:=πv−1(F)∩PofafacetFofPviseitherafacetofP
paralleltovora(d−2)-dimensionalfaceofP.InthelastcaseΓ→F
isanisomorphism,andthereexistsexactlyonefacetofPthatcontainsΓ
.vand

48

Chapter3.Reflexivepolytopes

7.Suppose−v∈P.ThenanyfacetofPeithercontainsv,or−v,oris
paralleltov,i.e.,afacetoftheformπv−1(F)∩PforF∈F(Pv).
8.(Pv)∗=∼P∗∩v⊥aslatticepolytopes.
PvisreflexiveifandonlyifP∗∩v⊥isalatticepolytope.
Proof.1.LetFbeafacetofPcontainingvandx.Ifλ>0,thenηF,x+λv=
−1−λ<−1,sox+λv∈P.HenceS⊆U.
Ontheotherhandletx∈U.ConsideringthepolytopeP∩lin(v,x)wesee
thereisafacetFofPnotparalleltovthatcontainsxwithηF,v<0.Since
Pisreflexive,wehaveηF,v=−1,hencev∈Fandx∼v.Thisimplies
.U=S2.Letm∈Pv∩Mv.Wehaveu:=ιv(m)∈U=S.Sothereexistsafacet
F∈F(P)withv,u∈F.By3.1.4(11)thereisaZ-basise1,...,ed−1,ed∗=vof
MsuchthatV(F)⊆{x∈MR:xd=1},i.e.,ηFisthedualvector−ed.Let
u=λ1e1+∙∙∙+λdedforλ1,...,λd∈R.Nowe1,...,ed−1isaZ-basisofMv.
Thereforeλ1,...,λd−1∈Z.Sinceu∈F,wegetλd=ud=−ηF,u=1,hence
.Mu∈3.TheprojectioninducesanhomeomorphismS→Pv,andhenceabijection
oundaries.btheirof4.Thefirststatementsfollowfromthefirstandthethirdpoint.Letz∈
V(P)∩∂vbesuchthat]v,z[iscontainedintherelativeinteriorofafacetofP.
ThenthereisonlyonefacetF∈F(P)thatcontainsvandz.Sincez∈V(F)
wecanchooseanaffinehyperplaneHthatintersectsFonlyinzandisparallel
tov.ForPv:=πv(P)andH:=πv(H)letx∈H∩Pv.Itremainstoshow
thatx=z:=πv(z).Soassumenot.Hintersectsπv(F)onlyinz.Therefore
ιv(y)∈Fforally∈]z,x].FinitenessofF(P)impliesthatziscontainedin
anotherfacet=Fcontainingv,acontradiction.
5.Thisisproveninthesamemannerasthethirdpoint.
6.Thefirststatementsfollowfromthethirdandthefourthpoint.
Forthesecondstatementletdim(Γ)=d−2.NowobservethatifΓ→
Fwerenotinjective,afacetcontainingΓnecessarilywouldbeparallelto
v,soitsimageafacetcontainingF,acontradiction.ThereforeΓ→Fis
anisomorphismofpolytopeswithrespecttotheiraffinespans.Wechoose
x∈relintF.Lety:=ιv(x)∈S∩Γ.Byassumptionalsoy∈relintΓ.Let
G∈F(P)withv,y∈G.ThenΓ⊆G,henceGisoneofthetwofacets
Γ.tainingcon7.Let−v∈P.AnyfacetF∈F(P)satiesfies−1≤ηF,v=−ηF,−v≤
1.Fromthisthestatementsfollow.
8.ChooseagainafacetFofPwithv∈F,andaZ-basise1,...,edof
⊥∼Mcosucordinateshthatweed=getv(Pvand)∗e=1,.{.y.,e∈d−R1dis−1a:Zy-basis,xof≥M−1∩η∀xF=∈PMvv}.=In{ythese∈
Rd−1:(y1,...,yd−1,0),(x1,...,xd)≥−1∀x∈P}=P∗∩v⊥.
Thelastpointanditsproofaretakenfrom[AKMS97,theoreminsection3].

3.2.Projectingalonglatticepointsontheboundary49
Let’sconsiderthealgebraic-geometricinterpretationoftheprojectionmap:
ThroughoutletX:=X(M,)for:=ΣPandP⊆MRreflexive.
Letv∈V(P),τ:=pos(v)∈(1),andπvasinthepreviousproposition.
Asin[Oda88,1.7]thefan:={πv(σ):σ∈,τ≤σ}={pos(πv(F)):F∈
F(P),v∈F}definestheprojectivetoricvarietyVτthatisthetorusinvariant
primedivisorcorrespondingtotherayτ.
OntheotherhandthereistheprojectedpolytopePv:=πv(P)thatspans
afanvin(Mv)R,wedenotethecorrespondingprojectivetoricvarietybyXv.
Inthefollowingwediscuss,howandwhenVτandXvarerelated.
WechooseatriangulationT:={Tk}of∂vintosimpliciallatticepolytopes.
ThenT:={pos(πv(Tk))}isasimplicialfanin(Mv)RwithcorrespondingQ-
factorialcompletetoricvarietyXT.FromProposition3.2.2(5)itfollowsthat
Tisacommonrefinementofandv.Especiallythereareinducedproper
birationalmorphismsXT→VτandXT→Xv.
Ingeneralisnotarefinementofv(andviceversa).Howeverinthecase
thatPissimplicial,wecanchooseTobviouslyinsuchawaythatT=,in
Inparticularorderthistodraimplieswthatconclusionsisabaoutrefinementhetofcanonicalv.divisorandsingularitiesof
theselower-dimensionaltoricvarietiesthereisthefollowingsufficientassump-
tion:∃f∈N>0:∀w∈V(P)∩∂v:|[v,w]∩M|−1=f.(3.1)
Supposethisconditionholds.Foranyw∈V(P)∩∂vthesecondpointinthe
propositionimplies|[0,πv(w)]∩Mv|−1=f.Furthermoreletw∈V(Pv),τ:=
pos(w).Sinceproposition3.2.2(4)impliesιv(w)∈V(P)∩∂v,theprevious
considerationyieldsvτ=(1/f)w,henceQv=(1/f)Pv(forthisnotationsee
p.37).Therefore−KXvisanampleQ-Cartierdivisor,i.e.,XvisatoricFano
.varietyDefinition3.2.3.X,respectivelyP,iscalledsemi-terminal,ifforallv∈V(P)
condition(3.1)holdsforf=1,i.e.,[v,w]∩M={v,w}forallw∈V(P)∩∂v.
Proposition3.2.4.LetP⊆MRbeareflexivepolytope.
1.Pissemi-terminaliffPvisaFanopolytopeforallv∈V(P).
2.PisterminaliffPvisacanonicalFanopolytopeforallv∈V(P).
Proof.1.Fromlefttoright:Thisholds,sinceQv=PvisaFanopolytope.
Fromrighttoleft:Letv=w∈V(P)withv∼w.ChooseCandF
asinProposition3.2.2(3)suchthatw∈V(C).ThereforeweseethatF:=
aff(πv(C))∩PvisafacetofPv.Henceby3.2.2(6)forΓ:=πv−1(F)∩Pwe
haveeitherΓ=GforafacetG∈F(P)thatisparalleltovorΓ=C.In
thefirstcaseηG,v=0andηG,w=−1,so|[v,w]∩M|=2.Inthesecond
caseobviouslyF=πv(C),soπv(w)∈V(F),hencebyassumptionaprimitive
latticepoint.From3.2.2(2)weget|[v,w]∩M|=2.
2.Fromlefttoright:Letv∈V(P),0=m∈Mv∩Pv.Inthenotationof
Proposition3.2.2(1,2)wegetιv(m)∈M∩∂P⊆V(P)byassumption.Hence
m=πv(ιv(m))∈∂PvbyProposition3.2.2(4).
Fromrighttoleft:Assumethereisaw∈∂P∩M,w∈V(P).Thenwisa
properconvexcombinationofverticesofPcontainedinacommonfacet.Letv
beoneofthem.Thenπv(w)isintheinteriorofPv,acontradiction.

50

Chapter3.Reflexivepolytopes

Corollary3.2.5.LetXbeaGorensteintoricFanovariety.
Thenthefollowingtwostatementsareequivalent:
singularitiesterminalhasX1.2.Xissemi-terminalandXvhascanonicalsingularitiesforanyv∈V(P)
Ifthisholds,thenXvisatoricFanovarietyforanyv∈V(P).
Inparticularweseethatterminalityisanecessaryconditionforobtaining
areflexivepolytopeunderprojection,howevernotsufficientford≥4.
Example3.2.6.Letd=4,ande1,...,e4aZ-basisofM.Wedefinethe
simplicialcentrallysymmetricreflexivepolytopeP:=conv(±(2e1+e2+e3+
e4),±e2,±e3,±e4).ThenPiscombinatoriallyacrosspolytope,has8vertices
and16facets.ItisaterminalFanopolytopebutnotasmoothFanopolytope,so
especiallyitadmitsnocrepantresolution.TheprojectionPe4alongthevertex
e4has6vertices,Pe4isevenaterminalFanopolytopebutnotreflexive.This
polytopeistakenfrom[Wir97]whereitisusedinadifferentcontext.
NowweconsiderVτ:ToensurethatthecanonicaldivisorofVτisQ-Cartier,
weneedingeneraltheQ-factorialityofX.SoletPbesimplicialandassume
againthatcondition(3.1)holds.Thenisacoherentcrepantrefinementof
v.Hence−KVτisanefQ-Cartierdivisor,i.e.,VτisatoricweakFanovariety.
FromCor.3.2.5weget:
Corollary3.2.7.LetXbeaQ-factorialGorensteintoricFanovariety.
Thenthefollowingtwostatementsareequivalent:
singularitiesterminalhasX1.2.Xissemi-terminalandVτhasterminalsingularitiesforanyτ∈(1)
Ifthisholds,thenVτisaQ-factorialtoricweakFanovarietyforanyτ∈(1).
FinallytoadditionallyderivetheGorensteinproperty,i.e.,thatthecanonical
divisorisZ-Cartier,weneedastrongerassumption,thatistrivialinthecase
ofasmoothFanopolytope:
ForanyF∈F(P)withv∈FandC∈F(F)withv∈C
thereexistw1,...,wd−1∈C∩Msuchthat(3.2)
w1,...,wd−1,visaZ-basisofM.
Ifthisconditionisfulfilled,then(3.1)holdsforf=1,PvisreflexivebyProp.
3.2.2(3),andXvisaGorensteintoricFanovariety.IfPisalsosimplicial,then
VτisaGorensteintoricweakFanovariety.
SupposenowXissemi-terminal,simplicialandVτisnonsingularforany
τ∈(1).Itfollowsfrom3.1.4(11)that(3.2)holdsforanyv∈V(P).ThenPv
isreflexive,inparticularcanonicalforanyv∈V(P),henceCor.3.2.5implies
thatPisterminal.SincePisalsosimplicial,theassumptionimpliesthatPis
alreadyasmoothFanopolytope.Wehaveproventhefollowingcorollary:

3.2.Projectingalonglatticepointsontheboundary51

Corollary3.2.8.LetXbeaQ-factorialGorensteintoricFanovariety.
Thenthefollowingtwostatementsareequivalent:
nonsingularisX1.2.Xissemi-terminalandVτisnonsingularforanyτ∈(1)
Ifthisholds,thenVτisaQ-factorialtoricweakFanovarietyforanyτ∈
(1),andXvisaGorensteintoricFanovarietyadmittingthecoherentcrepant
resolutionVτ→Xvforanyv∈V(P),τ=pos(v).
TheimportantfactthattheprojectionofasmoothFanopolytopeisreflexive
wasalreadyprovenbyBatyrevin[Bat99,Prop.2.4.4],howeverheusedthe
notionofaprimitiverelation[Bat91]andresultsofReidabouttheMoricone
[Rei83].AnotherproofthatisessentiallyequivalenttotheoneofBatyrevwas
donebyEvertz[Eve88]byexplicitlycalculatingequationsoffacets.
ThereisnowageneralizationofthisresulttotheclassoftoricFanovarieties
withlocallycompleteintersections.Thesevarietieswerethoroughlyinvestigated
in[DHZ01],whereitwasproventhattheyadmitcoherentcrepantresolutions.
Proposition3.2.9.LetXbeaGorensteintoricFanovarietythathassingu-
laritiesthatarelocallycompleteintersections.
Thenthefollowingthreestatementsareequivalent:
semi-terminalisX1.singularitiesterminalhasX2.3.AnyfacetofPcanbeembeddedasalatticepolytopeinto[0,1]d−1
Ifthisholds,thenXvisaGorensteintoricFanovarietyforanyv∈V(P).
IfadditionallyXisQ-factorial,thenXisnonsingular.
Proof.ThefacetsofthecorrespondingreflexivepolytopeParesocalledNaka-
jimapolytopes,acomprehensivedescriptioncanbefoundin[DHZ01].Using
theirresultsitisstraightforwardtoprovethefollowingstatementbyinduction
:nonLetvbeavertexofann-dimensionalNakajimapolytopeFinalatticeM
suchthat|[w,w]∩M|=2forallw,w∈V(P),w=w.ThenFisempty,can
beembeddedasalatticepolytopein[0,1]n,andforanyfacetC∈F(F)with
v∈Cthereexistnverticesw1,...,wnofCsuchthatw1−v,...,wn−visa
-basis.ZFromthisresultthepropositionisobvioususingcondition(3.2).

InExample3.2.6wehavethesituationthatanyfacetsatisfiesthethirdcon-
hasditionofterminaltheprevioussingularities,propbutosition.XvisThenevercorrespaondingGorensteinvariettoricyXFanoisQvariet-factorial,y,so
Xdoesnothavelocallycompleteintersections.

52Chapter3.Reflexivepolytopes
3.3Pairsoflatticepointsontheboundary
ThroughoutthesectionletPbead-dimensionalreflexivepolytopeinMR.
tionsInwere[Bat91]definedtheforimportantnonsingularnotionsproofjectiveprimitivetoriccvollearietiesctionsandandusedprimitivein[Bat99rela-]
fortheclassificationoffour-dimensionalsmoothFanopolytopes(foramore
generaldefinitionandtreatmentofthesenotionsseesection4.1).Hereprimi-
tiverelationsareinsomesenseminimalintegerrelationsamongtheverticesofa
smoothFanopolytope,hencetheygive1-cyclesonthevariety(seep.23),that
evengeneratetheMoricone(see[Rei83]and[Bat91]).Unfortunatelytheseuse-
fultheytoolsrelyoncannotthesimplyexistencebeoflatticegeneralizedbasestotheamongclassthevoferticesreflexivofeapfacet.olytopHoes,wevsinceer
thenextpropositionshowsthatinthesimplestyetmostimportantcaseofa
thatprimitivareenotconcollectiontainedofinorderatwcommono,i.e.,afacepair(seeofDef.lattice4.1.4),poinwtseonstillthehavbeaoundarykind
ofgeneralizedprimitiverelation(v+w=0orav+bw−z=0)that(unlikethe
trivialchoiceinDef.4.1.6)mimicsthegoodpropertiesofthenonsingularcase:
Proposition3.3.1.LetP⊆MRbeareflexivepolytope,v,w∈∂P∩M,v=w.
Exactlyoneofthefollowingthreestatementsholds:
wv1.∼2.v+w=0
3.v+w∈∂P
Letthethirdconditionbesatisfied.Thenthefollowingstatementshold:2
suchv,wthat,isaifZwe-basissetzof:=linz(v(,v,ww))∩:=M.avTher+ebw,existsthenzexactly∈∂P,onevp∼airz(a,andb)w∈∼N>z.0
over:eMori.a=1orb=1.a=|[w,z]∩M|−1andb=|[v,z]∩M|−1.
IfF∈F(P)withv,z∈F,thenηF,w=ab−1.
ii.Anyfacetcontainingz(orv+w)containsexactlyoneofthepointsvor
.wiii.ForanyF∈F(P)containingvandzthereexistsafacetG∈F(P)
containingwandzsuchthatF∩Gisa(d−2)-dimensionalfaceofP.
iv.Ifz∈V(P),b=1,and]v,z[iscontainedintherelativeinteriorofafacet
ofP,then[w,z]iscontainedinanedge.
Proof.Letv∼wandv+w=0.Thefirstconditionimpliesthatforanyfacet
F∈F(P)wehaveηF,v+w=ηF,v+ηF,w>−2.Howeverreflexivity
ofPimpliesthatthismustbeanaturalnumbergreaterorequalto−1,so
v+wSince∈Pconbyv(0,dualitv,wy.)isWeangetempt0=yvtw+w∈∂o-dimensionalP,bpecauseolytopPise,bycanonical.3.1.8(1)wesee
thatv,wisaZ-basis.
LetFbeafacetofPcontainingv+w.WemayassumeηF,v=−1and
ηF,w=0.Thisimpliesv∼v+w.

3.3.Pairsoflatticepointsontheboundary

53

Wecanusethisconsiderationagainforthepairv+w,w.SinceF∩Mis
finite,thiseventuallyyieldsanaturalnumberb∈N>0suchthatz=v+bw∈F
andw∼z.Inparticulara=1.Thisprovestheexistenceofzandi.
ii.LetF∈F(P)withz∈F.Assumev,w∈F,hence−1=ηF,z=
aηF,v+bηF,w≥0,acontradiction.
iii.LetF∈F(P)withv,z∈F.Sincez∈∂Fandzv,thereexistsa
faceH∈F(F)withz∈Handv∈H.Bythediamondproperty(p.23)there
existsafacetG∈F(P)suchthatG∩F=H.Inparticularz∈Gandv∈G,
sobythesecondpointw∈G.
iv.FollowsfromProp.3.2.2(4)appliedtoπv.
Thestatementandtheproofshallbeillustratedbythefollowingfigure:
z = v + 2 wvv+wwForanotherexamplenotethatthefollowingsituationcannotoccurfora
three-dimensionalreflexivepolytope,sinceconv(x,y,z(v,w))doesnotcontain
vorw,acontradictionto3.3.1(ii);thereadershouldnottrytoprovethisby
facets:calculatingexplicitlyvxz(v,w)ywThesymmetricrelation∼definesagraphW(P)on∂P∩M,where∂P∩M
aretheverticesofW(P),and{v,w}isanedgeofW(P)ifandonlyifv∼w.
Fromthepreviouspropositionwecannoweasilyderivethefollowingcorol-
laryaboutcombinatorialpropertiesofreflexivepolytopes:
Corollary3.3.2.LetP⊆MRbeareflexivepolytope.
1.Anypairofpointsin∂P∩Mcanbeconnectedbyatmostthreeedges
ofthegraphW(P),withequalityonlypossiblyoccuringforacentrally
symmetricpairofpoints.
2.ThepreviousstatementalsoholdsforthesubgraphofW(P)whosevertices
consistofV(P);thisisapurelycombinatorialobject.Inthecaseofa
simplicialpolytopethisisjusttheusualedge-graphontheverticesofP.
3.Bydualizingwegetthatapairoffacetsofareflexivepolytopeiseither
parallel,containsacommonvertex,ordoeshavemutuallynon-trivialin-
tersectionwithanotherfacet.
Proof.1.Letv,w∈∂P∩M,v=w,v∼w.Ifv+w=0,then3.3.1yields
zw:=∈zV((v,Pw),)w∈∂=Pw∩,Mwithsucwh∼thatw.vNo∼wz∼applyw.If3.3.1v+forw=the0,thenlatticewpeoincantscv,howose.

54

Chapter3.Reflexivepolytopes

z:=2.z(vLet,w)v,∈w∂∈PV∩(PM.),vNo=ww,v3.3.1(iii)∼w.Wimpliesecanthatagainthereassumeexistsvz+∈wV=(P0,)sucandh
thatv∼z∼w.
3.Thisisobviousfromthesecondpoint.

Thestatementandtheproofofthefirstpointisillustratedbythefollowing
figures:vwvzvvz’

w’w’ww=−vw=−vWithoutusingtheexistingclassificationoftwo-dimensionalreflexivepoly-
topesapplication(seeinProp.thecase3.4.1)ofdthe=2prop(forositiontheproandofofthethecorollarysecondpyieldointanuseimmediatestatement
osition).proptheofiCorollary3.3.3.LetPbeatwo-dimensionalreflexivepolytope.
1.Phasatmostsixvertices;equalityoccursiffPisoftype6ainProp.
3.4.1.2.AfivenyfaclatticeteofpPointsciffontainsPisatofmosttype8fivecinlatticPreop.points;3.4.1.thereexistsafacetwith

Thisfirstpointisalsoadirectconsequenceof[PR00,Thm.1]statingthat
|∂P∩M|+|∂P∗∩N|=12foratwo-dimensionalreflexivepolytope,where
howevernodirectcombinatorialproofisknownthatdoesnotusesomekindof
induction(seealsoProp.3.4.1below).Inhigherdimensionsthereisnosuch
directrelationbetweenthelatticepointsinthedualpairofreflexivepolytopes.
Anotherapplicationistoshowthatcertaincombinatorialtypesofpolytopes
cannotberealizedasreflexivepolytopes.Asanexamplewehavealookatthe
regularpolyhedra(seeforinstance[Sti01]).
Thesecontainthed-simplex,thed-cube,anditsdual,thed-crosspolytope;
ford=3thesearethetetrahedron,thecube,andtheoctahedron.Inanydimen-
sionthesecombinatorialtypescanberealizedasreflexivepolytopes.However
apartfromthesethreeinfinitefamiliestherearefivesporadiccases.Ford=3
wealsohavetheicosahedron(simplicial,with12verticesand20facets),and
itsdual,thedodecahedron.Moreoverford=4therearethe24-cell(self-dual,
with24vertices,facetsareoctahedra),the600-cell(simplicial,with120vertices
and600facets),anditsdual,the120-cell.Nowthereisthefollowingresult:
Corollary3.3.4.Thereisnoreflexivepolytopethatiscombinatoriallyisomor-
phictothedodecahedronortheicosahedron.Thereisnoreflexivepolytopethat
iscombinatoriallyisomorphictothe120-cellorthe600-cell.

3.4.Classificationresultsinlowdimensions

55

Proof.LetPbeareflexivepolytopeandd=3.BydualitywecanassumethatP
isyieldscombylobinatoriallyokingattheisomorphicedge-graphtoanthatPicosahedron.iscentrallyCorollarysymmetric.3.3.2(2)Howevimmediatelyerany
8verticesthree-dimensionalaswillbecenprotrallyveninsymmetricTheorem3.5.11.simplicialreflexivepolytopehasatmost
pairsofFinallyvberticesyofCorollarythesimplicial3.3.2(2)and600-celldualitthatyitiscannotenoughbetonoteconnectedthatbyattheremostare
threeedges(see[Sti01,Fig.5]).

Itisnowanastonishingobservation(see[KS02])thattheself-dual24-cellcan
beuniquelyrealizedasareflexivepolytopewithvertices{±ei:i=1,...,4}
∪{±(ei−ej):i=1,2,j=2,3,4,j>i}∪{±(ei−e3−e4):i=1,2}∪{±(e1+
e2−e3−e4)}fore1,...,e4aZ-basisofM.Itisevencentrallysymmetricand
terminal.Hereitisinterestingtonotethenecessityoftheseconditions:
binatorialCorollarylya3.3.5.24-celLl.etPThenbeaPhastobfour-dimensionalecentralrlyeflexivesymmetricpolytopandePthatterminal.iscom-
toProtheof.Letusualvbanetipaovdalertexpoinoft.P.NoAssumewchov+osewthe=v0.ertexItisw∈easyV(toP)seecorresp(see[Sti01onding,
Fig.4])thattheintersectionofafacetcontainingvandafacetcontainingwis
emptyorconsistsofauniquevertexzwhere]v,z[and]w,z[arecontainedinthe
relativeinteriorsofthesefacets.Thisimpliesv∼wandz(v,w)=z∈V(P),a
conThetradictionterminalittotheyoflastPpcanoinbteofprovProp.enin3.3.1.ananalogousway.

Duetothelist[KS04b]thepreviouslydescribedpolytopeiseventheonly
four-dimensionalreflexivepolytopewith24verticesand24facetssuchthatany
vertexiscontainedin6facets.

3.4Classificationresultsinlowdimensions
Inthissectionwewillgiveasurveyofspecialclassesofreflexivepolytopesand
previouslyachievedclassificationresultsinlowdimensions.
SmoothFanopolytopes,astheyformthemostimportantclassofreflexive
polytopes,wereintensivelystudiedoverthelastdecadebyBatyrev[Bat91,
Bat99],Casagrande[Cas03a,Cas03b],Debarre[Deb03],Sato[Sat00],etal.It
couldbeproventhatthereare18smoothFanopolytopesford=3(see[Bat82a,
WW82])and124ford=4(see[Bat99,Sat00])uptoisomorphism.Herewe
willhavealookatrecentclassificationresultsofreflexivepolytopesinlow
dimensions.Ford=1thepolytope[−1,1]correspondingtoP1istheonlyFanopoly-
tope.Ford=2anycanonicalFanopolytopeisreflexiveby3.1.8(1),andthese
isomorphismclassescanbeeasilyclassified(e.g.,see[KS97]or[Sat00,Thm.
6.22]).Fortheconvenienceofthereaderandlaterreferencewewillgivethelist
ofthe16isomorphismclassesofreflexivepolygonsaswellasasimpleproof.
Proposition3.4.1.Thereareexactly16isomorphismclassesoftwo-dimen-
sionalreflexivepolytopes(thenumberinthelabelsdenotesthenumberoflattice
pointsontheboundary):

56

3

5a

6c

4a

5b

6d

Chapter3.Reflexivepolytopes

4b

6a

7a

7b

4c

6b

98c8b8aThereexistthefollowingdualpairsofreflexivepolygons:
(3,9),(4a,8a),(4b,8b),(4c,8c),(5a,7a),(5b,7b).
Eachofthereflexivepolygons6a,6b,6c,6disisomorphictoitsdual.
Proof.LetPbeatwo-dimensionalreflexivepolytope.Wedistinguishthree
cases:tdifferen1.AnyfacetofPcontainsonlytwolatticepoints,i.e.,PisaterminalFano
polytope.Therearethreedifferentcases(seeProp.3.3.1):
(a)Piscombinatoriallyatriangle.
By3.1.8(1)wemayassumethat(1,0),(0,1)areverticesofP.Letx
bethethirdvertex.FromProp.3.2.4(2)itfollowsthattheprojection
ofPalong(1,0)isacanonicalFanopolytope,i.e.,isomorphicto
[−1,1],hencex2=−1.Byprojectingalong(0,1)wegetx1=−1,
soPisoftype3.
(b)Thereexistthreeverticesu,v,w∈V(P)withu+w=v.
SincePisaterminalFanopolytope,Prop.3.3.1impliesu∼vand
w∼v,andwemayassumeu=(−1,1),v=(0,1),w=(1,0).Again
projectingalongvyieldsP∩{(−1,x):x∈Z}⊆{(−1,0),(−1,1)},
P∩{(0,x):x∈Z}⊆{(0,−1),(0,0),(0,1)},P∩{(1,x):x∈Z}⊆
{(1,−1),(1,0)}.Wegetaspossibletypes4b,5a,6a.

3.4.Classificationresultsinlowdimensions57
(c)Anytwoverticesthatarenoneighboursarecentrallysymmetric.
ThisimmediatelyimpliesthatPisoftype4a.
2.ThereexistsafacetFcontainingexactlyonelatticepointinrelintF.
PWe⊆ma{xy∈Rassume2:−V1(F≤)x=1{≤(−11,,x1)2,≤(1,11)}.}.SinceThen(0b,y−1)Prop.isnot3.2.2(1)conwtainedehavine
2innexttP,tenwegetPisomorphism⊆{x∈typRes:4cx,52b≥,6b−,63c},6.dF,7arom,7b,8thisa,8bw,8ec.readilyderivethe
case.remainingThe3.Wwithemavy≤−assume1Vminimal.(F)=As{((1−,1,0)1)is,(a,not1)in}theforain∈teriorN,aof≥P,2.weLethavve∈vV(≤P0.)
12mThenustbbeyanassumptionequality,hencenecessarilyPisofcontypv((e−9.1,1),(−1,−2),(2,1))⊆P.This

TheproofincludesthestatementthatthereareexactlyfivetoricDelPezzo
spsurfacondsesto,Pi.e.,2,4tawtoP1×o-dimensionalP1,4btothenonsingularHierzebructorichFanosurface,v5arieties:atoPT1yp×eP13blocorre-wn
upinonetorusinvariantpoint,6atoP1×P1blownupintwotorusinvariant
points.Thisresultcanalsobeprovenbybirationalfactorisation[Oda88,Prop.
2.21],primitiverelations[Bat91]ordeterminants[Ewa96,Thm.V.8.2].
Ingeneralevenford=3therearetoomanyreflexivepolytopestogivea
classificationbypencilandpaper.Howeverbyrestrictingtosmallerclassesof
observreflexiveationspolytopandesitisclassificationstillinresultsterestingbytodirectlyfindrigoroususingtheirmathematicalintrinsicpropproofserties:of
RecentlyKasprzykclassifiedin[Kas03]all634three-dimensionalterminal
pusingolytopaesbycomputerfirstprogramdescribingforthetheminimalremainingcasesones.purelyIn[Kas04]mathematicallyacompleteandlistthenis
pavossibleailable,towhererecoverforalleach100polytopterminalethereflexiveGorensteinindexthree-dimensionalisspecified,polytopsoesthatit(noteis
thethatinGorenstein[Kas03,indexKas04]isterminalreferredFtoanoasptheolytopFanoesareindex).calledIndepFanoendenptlyolytopes,(alreadyand
in2002)theauthorhasobtainedthislist(seeThm.4.3.1),thisclassification
issymmetricexplainedinreflexivtheepnextolytopceshapter.(seeW[Wagnerag95]).classifiedMoreoverthethree-dimensionalauthorgaveacenclassi-trally
groupficationdoofesnothavthree-dimensionalenon-trivialreflexivfixpeoinptsolytop(seeesThm.where5.4.5).thelinearautomorphism
KreuzerandSkarkedescribedin[KS97,KS98,KS00]ageneralalgorithmto
classifyreflexivepolytopesinfixeddimensiond.Usingtheircomputerprogram
PALP(see[KS04a])theyappliedtheirmethodford≤4,andfound4319reflex-
ivepolytopesford=3and473800776ford=4.Theyalsodescribedhowtofind
anormalformoflatticepolytopes,toricfibrationsandsymmetries.Thecom-
thepletelistoffour-dimensionalthree-dimensionalreflexivepolytopreflexivesepcanolytopbeesfoundandonatheirsearcwhableebpagedatabase[KS04bof].
aHerereflexivethesimplexideasofantheirexplicitalgorithmapproachshallduebetoshortlyConradsskisetcheddescrib(inedtheincasesectionof
resp3.6):ectFirsttothetheysoshocalledwedthatIP-propanyerty,latticei.e.,itpconolytoptainseQthe⊆NRoriginthatinisitsinminimalterior,withcan

58Chapter3.Reflexivepolytopes
becombinatoriallydecomposedintosimplices,asocalledIPsimplexstructure.
Thereare2,3,9suchstructuresford=2,3,4.Foranysimplexwehaveaunique
weightsystem(seesection3.6),henceanIP-simplexstructureyieldsasocalled
cvombineerticesdofQweight,andMsystemits(CWS).dualLetNlattice.coarsestThenbethetheCWSlatticeqofQgeneratedbdeterminesythe
finestQ∗andhenceQuptoanisomorphismofMfinestrespectivelyNcoarsest.Now
wedefinethelatticepolytopeQ(q):=conv(Q∗∩Mfinest).Theyprovedthatif
Q(q)hastheIP-property,thenthisistrueforanyweightsystemintheCWS.
Ford≤4thisevenimpliesQ(q)tobereflexive.Nowtheydeterminedanysuch
possibleCWS,andhenceclassifiedanylatticepolytopeQthatisminimalwith
canrespectalwatoysthefindsucIP-prophaQert⊆y.NWhenwithwVe(Qha)ve⊆aV(P∗reflexiv).eHencepolytopP⊆eQP∗,⊆whereMR,thewe
R∗vjusterticeshaveoftoQchoosegeneratesomeMfinestlattice.NowsubpinolytoporderetoPrecoofvQer∗our(withreflexivrespecteptoolytopMfinestew).e
Thenwecheck,ifthevertex-pairing-matrixisintegral,andinthiscaseweagain
findMasaalatticeM⊆MfinestcontainingtheverticesofP.Thiscanbe
doneinasystematicwayusingHermitenormalforms(seeThm.3.6.6)
spectivUsingelythislistwfour-dimensional,efind194,respsimplicialectivelyreflexiv5450,epclassesolytopofes.Moreovthree-dimensional,erthisyieldsre-
151four-dimensionalterminalsimplicialreflexivepolytopes(ford=3anysuch
polytopeissmooth,see3.1.8(2)).
3.5Sharpboundsonthenumberofvertices
ThroughoutthesectionletPbead-dimensionalreflexivepolytopeinMR.
Sinceinhigherdimensionsonlyinveryspecialcasesclassificationresults
exist,onetriestofindatleastsharpboundsoninvariantsandtocharacterizethe
caseofequality.Hereweexaminethenumberofverticesofareflexivepolytope.
Whileforgeneralreflexivepolytopesonlyaconjectureduetotheauthorexists,
inthesimplicialcasewehaveanalmostcompleteanswer.Thiswasachievedby
Casagrandeafterthepublicationofthepreprint[Nil04a].Moreovertheauthor
couldgeneralizepreviousnon-trivialresultsofDebarreandCasagrandeabout
smoothFanopolytopestosimplicialreflexivepolytopes.
Thenumberofverticescorrespondstotherankoftheclassgroupofthe
toricvarietyX:=X(M,ΣP)associatedtothefanspanned,wehaveby(1.1)
rankCl(X)=|V(P)|−d.(3.3)
ThecomputerclassificationofKreuzerandSkarke(see[KS04b])yieldsthat
themaximalnumberofverticesofad-dimensionalreflexivepolytopeis6for
d=2,14ford=3and36ford=4(notethatthereisamisprintonpage1220
of[KS00]stating33asthemaximalnumberofvertices).
Definition3.5.1.Zd:=conv([0,1]d,−[0,1]d)iscalledthen-dimensionalstan-
dardlatticezonotope.Itisacentrallysymmetricterminalreflexivepolytope,for
d≥3itisnotsimplicial(see[DHZ01,ProofofThm.3.21]whereZ(d)=Zd−1).
ApictureofZcanbefoundonp.118.
HoweverZ23=conv(±[0,1]2)isthe(uptoisomorphism)uniquecentrally
symmetricself-dualsmoothFanopolytopewith6vertices(oftype6ainProp.
3.4.1).WedenotebyS3:=X(M,Σ2Z2)theassociatednonsingulartoricDel
PezzosurfacethatistheblowupofPinthreetorus-invariantpoints.

3.5.Sharpboundsonthenumberofvertices

59

Conjecture3.5.2.LetPbead-dimensionalreflexivepolytope.Then
|V(P)|≤62d,
whereequalityoccursifandonlyifdisevenandP∼=(Z2)2d.
Remark3.5.3.Itwouldbeenoughtoprovethisconjecturefordeven,because
productsofreflexivepolytopesareagainreflexive.
Ford=2arigorousproofisknown(seeCor.3.3.3(1)andProp.3.5.5(1)
below).Ford≤4theauthorcheckedthisconjectureusingtheclassificationdata
[KS04b]andthecomputerprogramPALP(see[KS04a]).Inhigherdimensions
thisconjecturewillbeprovenforcentrallysymmetricsimplereflexivepolytopes
inthelastchapter,seeTheorem6.2.2onpage149.
Remark3.5.4.Inthepaper[HM04]Haaseintroducesthenotionofthere-
flexivedimensionrefldim(Q)ofalatticepolytopeQastheleastdimensiond
areflexivepolytopePcanhavethatcontainsQasaface.Heprovesthat
refldim(Q)isfinite.Assumingthepreviousconjectureweget
refldim(Q)>2log6V(Q).
Ifdisodd,thereisnosuchobviouscandidateasZd/2isfortheevencase.For
d=3themaximalnumberofverticesis14,andZ3istheonlyreflexivepolytope
withthisn(dum−3)b/er2ofvertices,(d−it3)is/2notsimple.Foroddd≥5thereflexive
polytopeS×Z3has614vertices.Ofcourseonecouldconjecture
thatthisw3erethemaximalnumberofvertices.LetPhavethemaximalnumber
ofverticesfordodd,thenwegetassumingthepreviousconjecture
6(d−1)/214≤|V(P)|<6(d−1)/2√6=6d/2.
6Ford=5,thiswouldimply84≤|V(P)|≤88.Soevenassumingthecorrectness
ofaboveconjecturetherecouldexistareflexivepolytopewithmorethan84
verticeswithoutimplyinganobviouscontradiction.
Thenextresultyieldstwocoarseupperboundsonthenumberofverticesof
areflexivepolytopeintermsofsomecombinatorialinvariantsofthefacets.The
firstboundisastraightforwardgeneralizationofaboundduetoVoskresenskij
andKlyachko[VK85,Thm.1]originallyproveninthesettingofasmoothFano
polytope.Thesecondupperboundisageneralizationof[Deb03,Thm.8],
whereDebarreimprovedfromaboundoforderO(d2)onthenumberofvertices
ofasmoothFanopolytopetoaboundoforderO(d3/2).Werecovertheoriginal
resultsforsimplicialreflexivepolytopes.Ofcoursebydualizingonecanderive
upperboundsonthenumberoffacets.Theseresultswillbefurtherimproved
intheremainderofthissection(seeTheorem3.5.12andCorollary3.5.13).
Proposition3.5.5.LetPbeareflexivepolytope.
Defineα:=max(V(F):F∈F(P))andβ:=max(F(F):F∈F(P)).
1.|V(P)|≤2dα.
Morepreciselywedistinguishtwocases:
Ifα≥2d−3,then|V(P)|≤2d(α−d+2)−2.

60

Chapter3.Reflexivepolytopes

Ifα≤2d−3,then|V(P)|≤dα+α−d+1.
IfPissimplicial,i.e.,α=d,andd≥3,thisyields
|V(P)|≤d2+1.
2.|V(P)|≤(α−d+1)β+2+2(α−1)(d+1)((α−1)+(α−d+1)β)).
IfPissimplicial,i.e.,α=d=β,thisyields
|V(P)|≤d+2+2(d2−1)(2d−1).
Proof.AnalyzingtheproofsofThm.1in[VK85]andThm.8in[Deb03]inthe
invmoreariantsgeneralαandsettingβintoofaaccounreflexivtweepjustolytophave,ewtoeseereprovthatebyremarktaking5(2)theinsectiongeneral
2.3of[Deb03],becauseonlythereexplicitlyalatticebasiswasused.Thatresult
isessentiallythefirstpartofthenextlemma.

Lemma3.5.6.LetP⊆MRbeareflexivepolytope.
LetF∈F(P),u:=ηF∈V(P∗)and{Fi}i∈IthefacetsthatintersectFin
a(d−2)-dimensionalface.Letm∈∂P∩Mwithu,m=0.
Thenm∈∪i∈IFi.
LetadditionallyFbeasimplexwithV(F)={e1,...,ed}.Lete1∗,...,ed∗be
thedualR-basisofNR.Fori=1,...,ddenotebyFithefacetofPsuchthat
Fi∩F=conv(ej:j=i).
1.Fori∈{1,...,d}wehave
m∈Fi⇐⇒ei∗,m≥0.
2.Ifthereexistsi∈{1,...,d}suchthatm∈Fiandm∈Fjforallj∈
{1,...,d},j=i,thenm∼ei.
3.Assumethatforeveryi=1,...,d−1thereexistsalatticepointmion
Fisuchthatu,mi=0andei∗,mi=−1.Thene1,...,edisaZ-basis
.MofProof.ThefirstpartfollowsfromProp.3.2.2(6)forπm.
NowletFbeasimplex.Thenu=jd=1(−ej∗)∈N.Leti∈{1,...,d}.
Sincemi∈Fand0isintheinteriorofP,thenumberαi:=−1e∗−,mu,mii>0is
well-defined.WegetηFi=u+αiei∗.Fromthis1.isreadilyderivied.2.isjust
acorollary.In3.wegetαi=1andei∗=ηFi−u∈Nfori=1,...,d−1and
ed∗=−u−e1∗−...−ed∗−1∈N.
Thisproofisinspiredbyremark5(2)insection2.3of[Deb03].

Inthefollowingwewillfocusontheclassofsimplicialreflexivepolytopes,
i.e.,wherethecorrespondingvarietiesareQ-factorial,orequivalently,theclass
numberequalsthePicardnumber.Thepreviouspropositionalreadygavea
hintthatsimplicialreflexivepolytopesareactuallyquiteclosetosmoothFano
polytopesatleastwhenconsideringonlythenumberofvertices.Thismotivated
theauthortostatethefollowingexplicitconjecture:

3.5.Sharpboundsonthenumberofvertices

61

Conjecture3.5.7(Nill5/2004).LetPbead-dimensionalsimplicialreflexive
polytope.Then3d,deven,
|V(P)|≤3d−1,dodd.
FordevenequalityholdsifandonlyifP∗=∼(Z2)2d,i.e.,X=∼(S3)2d.
Remark3.5.8.Itwouldbeenoughtoprovethisconjecturefordeven:Assume
therewerea∗simplicial∗∼reflexivdepolytopePwithdoddand|V(P)|≥3d.Then
necessarilyP×P=(Z2),thiswouldimplyPtobecentrallysymmetricwith
|V(P)|=3d,acontradictiontododd.
berFofromvertices[Kas03a]wegetthree-dimensionalthatreflexivitysimplicialisessential,terminalbFecauseanoptheolytopmaximalecannhaum-ve
10.isThepreviousconjecturewasoriginallyproposedinthecaseofsmoothFano
polytopesbyBatyrev[Ewa96,p.337],andwasupto2003onlyrigorouslyproven
toholdfor(upto)five-dimensionalsmoothFanopolytopesbyCasagrandein
3.2].Thm.,[Cas03aTheboundisalsosharpintheodd-dimensionalcase,takeX=P1×(S3)d2−1.
Howeverevenford=3thereisexactlyoneanothersimplicialreflexivepolytope
with8vertices,itisasmoothFanopolytope,notcentrallysymmetric,andthe
correspondingtoricvarietyXisanequivariantS3-fibrebundleoverP1.Soitis
alsotemptingtoformulateanexplicitconjecture:
Conjecture3.5.9.InConjecture3.5.7equalityholdsfordoddifandonlyif
thed−1reflexivepolytopedefinesatoricvarietythatisa(nonsingular)equivariant
(S3)2-fibrebundleoverP1.
eralThisformwhoaswproevervenitincouldtheonlybnonsingulareconfirmedcasebyintheCasagrandecaseofaforcend≤trally5,inthissymmetricgen-
pairoffacets,thiswillbeproveninTheorem6.2.4.
Anotherobservationford≤4isthatthemaximalnumberoffacetsasim-
plicialreflexivepolytopecanhave(d=2:6,d=3:12,d=4:36)isachieved
bytheoneswiththemaximalnumberofvertices.Inevendimensionthisisof
coursejustacorollaryofconjectures3.5.2and3.5.7.
Remark3.5.10.Fortheproofoftheseconjecturesinlowdimensionsthe
Dehn-Sommervilleequationswereusedtogetherwiththefollowingtheoremby
Batyrev[Bat99,Prop.2.3.7]fornonsingulartoricFanovarieties,wherefiis
thenumberofi-dimensionalfacesofthecorrespondingsmoothFanopolytope:
12fd−3≥(3d−4)fd−2.
Itisastonishingtoobservethatford≤4bytheclassificationofKreuzerand
Skarkethisrelationisalsovalidforsimplicialreflexivepolytopes.Howeverthere
isnotyetanalgebraic-geometricexplanationforthisphenomenon!
ThemaingoalofthissectionistogiveaproofofConjecture3.5.7inthe
caseofadditionalsymmetriesofthepolytope:
Theorem3.5.11.Conjecture3.5.7holdsinthecaseofasimplicialreflexive
polytopePwhereP∗containsavertexu∈V(P∗)suchthat−u∈P∗.

62

Chapter3.Reflexivepolytopes

Theresultsinthissectionwerepublishedinthepreprint[Nil04a]inMay
2004.HoweverinOctober2004CasagrandeprovedConjecture3.5.7forthe
classofsmoothFanopolytopes.Forthissheneededthefactthatthesum
ofallverticesinthedualpolytopeiszero,sheprovedthisusingbirational
factorization.Thensheappliedasimpleandneatenumeratingargumenttoget
theupperbound,fortheproofsheonlyusedthepreviouslemmainthecaseof
asmoothFanopolytopeasgivenin[Deb03].Fortheequalitycaseshereferedto
Theorem3.5.11oftheauthor.InNovember2004theauthorpointedouttoher
thatthefirstfactisactuallyafolkloreresultinconvexgeometry(seeLemma
5.3.8)andcanforinstancebefoundinanotherpreprint[Nil04b]oftheauthor
publishedinJuly2004.InDecember2004Casagrandepublishedanewversion
ofherpaper[Cas04]whereshesimplifiedherapproachtogetthefollowingfinal
Thm.1(i)]:,[Cas04resultTheorem3.5.12(Casagrande12/2004).Conjecture3.5.7holds.
NotonlydoesthisresultrenderthecoarseboundsinProp.3.5.5obsoletein
thesimplicialcase,butanalyzingtheproofin[Cas04]itispossibletogetalso
animprovementofthegeneralboundinProp.3.5.5(2):
Corollary3.5.13.LetPbeareflexivepolytope.
Defineα:=max(V(F):F∈F(P))andβ:=max(F(F):F∈F(P)).
|V(P)|≤2α+(α−d+1)β.
UsingtheideasofCasagrande’sprooftheauthorrealizedthatTheorem
3.5.11couldactuallybereducedtothecentrallysymmetriccase,wherean
easierproofispossible,evenmore,acompleteclassificationisnowavailable,
seesection6.3.Howeverintheremainderofthissectionwegivetheoriginal
proofaspublishedin[Nil04a],sinceitusesatechnique(Lemma3.5.15)thatis
itself.interestinginThemainresultforanalyzingsmoothFanopolytopesisatheoremofReid
aboutextremalraysoftheMoriconeandprimitiverelations(see[Rei83]and
[Cas03a,Thm.1.3]).Althoughforsimplicialreflexivepolytopesthereisno
generalnotionofaprimitiverelation,forthesimplestcaseasdefinedinProp.
3.3.1westillhaveananalogousresult(recallDefinition3.2.1):
Lemma3.5.14.LetPbeasimplicialreflexivepolytope.
Letv∈V(P),w∈∂P∩Mwithv+w∈∂Pandz:=z(v,w).
Letx∈∂P,x∈{v,w,z},withx∼zandxawayfromv.
Thenconv(x,z,w)iscontainedinaface.
Moreoverexactlyoneofthefollowingtwoconditionsholds:
1.Anyfacetcontainingxandzcontainsalsow.
2.ThereexistsafacetFwithx,v,z∈F.
Thesecondcasemustoccur,ifw∈V(P)andxisawayfromw.
Ifthesecondcaseoccurs,wehave:
ForanysuchFthereexistsauniquefacetGwithx,w,z∈Gsuchthat
F∩Gisa(d−2)-dimensionalfaceofP.F∩GconsistsofthoseelementsofF
thatareawayfromv,respectivelythoseelementsofGthatareawayfromthe
(unique)vertexnotinF.Obviouslyw∈Fandv∈G.

3.5.Sharpboundsonthenumberofvertices63
Proof.Assumethefirstcasedoesnothold.Prop.3.3.1(ii)impliesthatthere
existsafacetF∈F(P)withx,z,v∈F.By3.3.1(iii)thereisafacetG∈F(P)
containingwandzsuchthatF∩Gisa(d−2)-dimensionalface.SinceF,G
aresimplicesandv∈V(F),theremainingstatementsarenowstraightforward.
Thetwocasesareillustratedinthefollowingfigureforathree-dimensional
reflexivepolytope(wherexasinthefirst,xasinthesecondcase):
vzx’v+wxwThenextresultisageneralizationofalemmaprovenbyCasagrande[Cas03a,
Lemma2.3]forsmoothFanopolytopes,herewerecovertheoriginalstatement
inthemoregeneralsettingofaterminalsimplicialreflexivepolytope.
Lemma3.5.15.LetPbeasimplicialreflexivepolytope.
Letv,w∈V(P),w∈∂P∩Mawayfromw.Furthermoreletv+w∈∂P
andv+w∈∂P,z:=z(v,w)andz:=z(v,w).
WedefineK:=P∩lin(v,w,w).ThenKisatwo-dimensionalreflexive
polytope(ofpossibletypes5a,6a,6b,7ainProp.3.4.1).
IfKisterminal,thenz=v+w,w=−v−w=−z,z=v+w=−w;and
either∂K∩M={v,w,z,w,z}or∂K∩M={v,w,z,w,z,−v=w+w=
z(w,w)}.
Essentiallythelemmastatesthatthefollowingfigureistwo-dimensional:
vz’zw’wProof.Letz=av+bwandz=av+bwasin3.3.1.Wenotethatwandz
areawayfromvandw;zisawayfromvandw.
Assumew∼z.Sincewisawayfromw∈V(P),itfollowsfrom3.5.14that
thereexistsafacetthatcontainsw,z,v,hencew∼v,acontradiction.
Thusw∼z;inparticular,z=z.Therearenowtwodifferentcases,and
itmustbeshownthatthesecondonecannotoccur.
1.v,w,warelinearlydependent.
By3.3.1therearethreepossibilities:
Ifw∼w,thenK=conv(v,z,w,z,w).Ifw+w=0,thenv∈conv(v+
w,v+w),acontradiction.Ifw+w∈∂P,thenK=conv(v,z,w,z,w,
z(w,w)).

64

Chapter3.Reflexivepolytopes

ThusinanycaseKisalatticepolytopewith5or6vertices,canonical,
henceby3.1.8(1)reflexive.Byanalyzingthecasesin3.4.1wegetthe
ts.statemenremaining2.v,w,warelinearlyindependent.
Hencealsoz,z,varelinearlyindependent.
Assumez∼z.3.5.14impliesthatconv(z,z,w)iscontainedinafacet
F∈F(P).Sincev∈Fandz∈F,3.3.1(ii)impliesw∈F,acontradic-
tiontow∼z.
Thusz∼z.
Byassumptionz+z=0,hencez+z∈∂P.Lety:=z(z,z)=
kz+lz∈∂P∩M.Wehavey∈{z,z,v,w,w},becausev,w,ware
t.endenindeplinearlyChoosey∈∂Pwithy=v+λ(y−v)forλ≥1maximal,sothatyisaway
fromv.Furthermorey∼zandy∼z.Soby3.5.14thereexistfacets
F1,F2∈F(P)suchthatconv(y,z,w)⊆F1andconv(y,z,w)⊆F2;
v∈F1,F2.
Nowchoosey=w+µ(y−w)∈Pforµ≥1maximal;soyisaway
fromw.Furthermoreconv(y,y,z,w)⊆F1andyawayfromv,soby
3.5.14thereexistsafacetG∈F(P)thatcontainsy,v,zandinb−1tersects
F1ina(da−−12)-dimensionalface.HencenecessarilyηF1,v=aand
ηG,w=b.
Kisathree-dimensionalpolytope.AnyfaceofKiscontainedinafaceof
P.Sincey,z,w(resp.y,z,w)arelinearlyindependent,F1∩K(resp.
F2∩K)isafacetofK.MoreoverF1∩K=F2∩K,becausew∼z.So
C:=F1∩F2∩KisavertexoredgeofKcontainingy.Sincealsow+z=0
andw∼z,wegetw+z∈∂P.Wesetx:=z(w,z)∈∂P∩M.Since
z,w,w(resp.z,w,y)arelinearlyindependent,wehavex=w(resp.
).y=xWedistinguishseveralcases:
(a)y=y.
IntheR-basisw,w,zofR3weseethatyhasnegativefirstandnon-
negativesecondandthirdcoordinate,sopos(w,z)∩[w,y]consists
ofonepointx.Wehavex∈]w,y[⊆F1.Moreoversincex∼z,we
getx∈]z,x].
i.w∈F2.
TheverticesofCconsistofyandw.Sincex∈C,wehave
x∼w,hencex=x∈]w,y[⊆C,thusalsox=y.Ifa=1,
itwere0=ηG,w>ηG,x>ηG,y=−1,acontradiction.
Thereforea≥2,b=1,ηG,w=a−1>0.
ThenηF2,v=ba−1and−1=ηF2,y=−k+laba−1−lb,thus,
1,=bsincek−1=ab−1−1∈N.(3.4)
alSinceb−1=0,b≥2anda=1.
Ifk=1,athen(3.4)yields1=a(b−1)≥a≥2,acontradiction.

3.5.Sharpboundsonthenumberofvertices65
Ifl=1,thenηG,y=−k−1+kbηG,w.So−1=ηG,y=
(1−µ)(a−1)+µηG,y−=a(1−µ)(a−1)−µk−µ+µkbηG,w.
ThisimpliesηG,w=µk+bk+a.Since(3.4)yieldsk=a(b−1),
thisimplies−1+1
ηG,w=bµb−1∈N.
Ontheotherhandµ≥1andb≥2yields0<µb−1+1<1,this
equation.previousthetradictscon.Fwii.∈This2immediatelyimpliesy=y.Assumex=x.Thisyields
x∈F1.Letx∈F1awayfromwsuchthatx∈[w,x].By
assumptionw,z,x,y,x,xarecontainedinF1andaff(z,y)∩
F1=F1∩G∩K.Now3.5.14yieldsx∈F1∩G,soy∈]z,x[.
Howeverw∼y,sow∼z,acontradiction.Hencex=x∈
[.y,w]Sincew∼xthereexistsafacetH∈F(P)containingw,w,y;
H⊇[furthermorew,y]andH=F1F∩2,Hsince=[ww,y∈]ofF2.K.HencethereareedgesF2∩
Sincew∼zwecandefineinadoublerecursionxr0:=x,
xl0:=x(z,w),xri:=x(xli−1,z),xli:=x(z,xri−1)fori∈N,
i≥1.Asw,y,zandw,y,zarelinearlyindependent,weeas-
ilyseethatthisprocedureiswell-defined,andxl0,xl1,xl2,...are
pairwisedifferentlatticepointsin]w,y[andxr0,xr1,xr2,...are
pairwisedifferentlatticepointsin]w,y[.Hencewehavecon-
structedinfinitelymanylatticepointsinP,acontradiction.
(b)y=y.
Ify=y,thenyisalatticepointintheinteriorofK,acontradic-
tion.Thusy=y.Thisimpliesy∈]v,y[,soy∈G,andconv(z,y,y,v)
iscontainedinafacetF∈F(P).3.5.14impliesthatthereexistsa
uniquefacetG∈F(P)thatcontainsw,z,ysuchthatF∩Gisa
face.2)-dimensionald(−Furthermoreba−1=ηF1,v>ηF1,y>ηF1,y=−1,henceb≥2
anda=1.EspeciallywegetηG,w=0.
i.w∈G.
Sincew∈V(P),w∈Fandwisawayfromw∈V(G),3.5.14
impliesthatw∈G∩F,acontradiction.
ii.w∈G.
LetC:=G∩F1∩K.ThenyisavertexofC.
AssumeCwereanedge.Letv∈V(C)withv=y.This
impliesv=w.Thenvisawayfromw,henceby3.5.14v∈G,
thereforev=y,acontradiction.
SoC={y},andthesamewayweseethat[w,y]isanedgeof
.KObviouslyx∈[w,y]∪[y,z],howeverbecausex∼z,thisyields
x∈]w,y].Since0=ηG,w>ηG,x≥ηG,y=−1,this
impliesx=y.

66

Chapter3.Reflexivepolytopes

Furthermorewehavey=(1−λ)v+λy=((1−λ)+λ(ka+
la))v+λkbw+λlbw,whereλ>1.Ontheotherhandx=
rw+Comparingsz=rthew+cosavefficien+tssbwforforwr,sand∈Nw,r,thiss≥yields1.
λl=s,λkb=r.
Fimpliesromthe1=firstr=λkbequation>kbwe≥get1,as=conλl>tradiction.l≥1,sos≥2.This

straighUsingtforwProp.ardtopro3.2.2(1-4)veaandcorollaryanalyzingofthetheppreviousossiblelemma:casesinProp.3.4.1itis
Corollary3.5.16.LetPbeasimplicialreflexivepolytopeandv∈V(P).
that−Therve∈Var(ePat).FmostorPthrve:=eπvvertic(P)esofandPMvnot:=inMthe/Zvstarwesethaveofv;equalityimplies
|V(P)|≤|∂Pv∩Mv|+4,
whereequalityimplies−v∈V(P).Therearenowtwocases:
1.Letw∈V(P)withw=−vandw∼v.
theThenstaranysetoflatticwepbutointnotonawaythebfromoundaryworofinPlinis(v,inw)the.Thisstarsetimpliesofvorin
|Pv∩Mv|+|intPw∩Mw|≤|∂P∩M|≤|Pv∩Mv|+|intPw∩Mw|+2;
ifthesecondequalityholds,then−v∈P.
2.Nosuchwasin1.exists.Then:
|V(P)|≤|∂Pv∩Mv|+2.
provenGoingbybackCasagrandetoalgebraicinthegeometrynonsingularwecasederivea[Cas03a,generalizationThm.2.4]:ofatheorem
Corollary3.5.17.IfXisaQ-factorialGorensteintoricFanovarietywith
torus-invariantprimedivisorVτ,thenthePicardnumberssatisfytheinequality
ρX−ρVτ≤3.

FinallyusingLemmas3.5.6and3.5.15wearenowreadytoprovethemain
theorem.Proexistsofaofvtheertexoremu∈V3.5.11.(P∗)LetwithP−beua∈P∗.simplicialLetFbreflexivetheepfacetolytopecorrespsuchondingthattothereu
andFthefacedefinedby−u.Nowdefinetheset{v1,...,vd}ofverticesnot
inFbutinfacetsintersectingFinacodimensiontwoface.

3.5.Sharpboundsonthenumberofvertices

67

Lemma3.5.6immediatelyimpliesthatV(P)\(V(F)∪V(F))={v∈V(P):
u,v=0}⊆{v1,...,vd}.Thisyieldsthebound|V(P)|≤3d.
InordertoproveConjecture3.5.7andtherebyfinishtheproofofTheorem
3.5.11wemayassumethat|V(P)|=3danddisevenbyRemark3.5.8.
Lete1,...,edbetheverticesofthefacetFsuchthatFi:=conv(vi,ej:
j=i)isafacetfori=1,...,d.Sinceby3.5.6anylatticepointxin∂Pwith
u,xi=0iscontainediinsomeFi,howeverviistheonlyivertexofFiwith
u,v≥0(resp.u,v=0),thisnecessarilyimpliesx=v.Henceweget
{x∈∂P∩M:u,x=0}={v1,...,vd}.
Analogouslyletb1,...,bdbetheverticesofFsuchthatconv(vi,bj:j=i)
isafacetfori=1,...,d.Thenwegetthefollowingthreefactsfori,j,k∈
{1,...,d}:
∙Fact1:Fori=j:vi∼vjorvi+vj=0.
(Proof:Assumenot.Thenthereexistsaksuchthatvi+vj=vk∈Fk.
By3.3.1(ii)thisimpliesvi∈Fkorvj∈Fk,givingvi=vkorvj=vk,a
tradiction.)con∙Fact2:Fori:ei+vi∈∂Pandbi+vi∈∂P.
(Proof:Sincevi∈Fjforallj=iandvi∈Fi,3.5.6(2)yieldsei∼vi.By
symmetrythesameholdsforbi.)
∙Fact3:Leti,jsuchthatei+bj∈∂P.Thenz(ei,bj)=ei+bj=vkfor
somei=k=j.
(Proof:Sinceu,ei+bj=0,letei+bj=vkforsomek.Assumeei∼vk.
By3.3.1thenalso2ei+bj∈∂P.Thisimpliesvk=1/2(bj+(2ei+bj)),a
contradictiontovk∈V(P).Bysymmetrywegetei∼vk∼bj.Byfact2
necessarilyi=k=j.)
Leti∈{1,...,d}.Byfact2wecanapplyLemma3.5.15tothevertices
vi,ei,bi.Fromfact3andanalyzingthepossibletypesin3.4.1wegetthat
P∩lin(vi,ei,bi)mustbeaterminaltwo-dimensionalreflexivepolytope,so−ei=
vi+bi∈F,−bi=vi+ei∈F,vi=z(−ei,−bi)=−ei+(−bi).Asthisistrue
foralli=1,...,d,wegetF=−Fand−ei,−bi∈V(P).
mapaesgivThiss:{1,...,d}→{1,...,d},i→s(i),suchthatbs(i):=−ei.
1.sisinjective,henceapermutation.
2.Therearenofixpointsunders,i.e.,s(i)=iforall{1,...,d}.
3.−vi=ei+bi∈∂Pforalli∈{1,...,d}.
(Proof:By3.5.15itisenoughtoshowthat−vi∈P.Assumenot.Fact
1iimpliesthenvi∼vs(i)=z(−es(i),−bs(i))=z(−es(i),ei),soby3.5.14
v∼ei,acontradictiontofact2.)
4.s◦s=id.
(Proof:Assumethereexistsani∈{1,...,d}suchthatforj:=s(i)we
havebs(j)=bi.Thisimpliesbi∼vs(i)=z(−es(i),−bs(i))=z(bs(j),ei),so
byassumptionand3.5.14bi∼ei.Thisisacontradictionto(c).)

68

Chapter3.Reflexivepolytopes

Property(d)impliesthatPiscentrallysymmetric.Furthermoresisa
productof2ddisjointtranspositionsinthesymmetricgroupof{1,...,d}.This
permutationsandtheset{e1,...,ed}ofverticesofFuniquelydetermineP,
becauseF=−Fandvi=−ei+es(i)foralli∈{1,...,d}.
Foranyi∈{1,...,d}wegetu,vi=0andei∗,vi=ei∗,−ei+es(i)=−1.
Hence3.5.6(3)impliesthate1,...,edisaZ-basisofM.Thisimmediatelyyields
theuniquenessofPuptoisomorphismofthelattice.

simpliceseReflexiv3.6Alatticepolytopeisdeterminedbytherelationsamongtheverticesandthe
coordinatesofsomelinearlyindependentfamilyofvertices.Oftenthefirstinfor-
mationcanbederivedfromthecombinatorialdataandthesecondinformation
isencodedinsomematrixnormalforms.Inthecaseofcentrallysymmetric
simplicialreflexivepolytopesthisisdescribedandillustratedinthelastchap-
ter.Themostgeneralapproachtodescribethecombinatoricsistousethenotion
oftheweightsystemofapolytope,especiallyprominentinthegeneralapproach
ofKreuzerandSkarke[KS97](seep.57).Hereweexaminethesimplestcase,
thatis,thecaseofareflexivesimplexindetail.Itiswell-knownthatthereisa
directrelationtoelementarynumbertheoryinthissetting.
Inthefirstsubsectionwesummarizeinanewandunifyingwaytheresults
ofConradsin[Con02](baseduponobservationsofBatyrevin[Bat94])about
latticesimplicesandtheirweightsystems.Hereweightedprojectivespaces
withGorensteinsingularitiescorresponduniquelytosocalledreflexiveweight
.systemsInthesecondsubsectionwegivethecorrespondenceofreflexiveweightsys-
temsandunitpartitions,thatis,unitfractionsthatsumuptoone.Usingnew
number-theoreticresultswegetasthemainresultupperboundsonthetotal
weightofweightsystemsofreflexivesimplices.
3.6.1Weightsystemsofsimplices
ThissubsectionsummarizesresultsofBatyrevin[Bat94,5.4,5.5]andConrads
].[Con02inThefollowingnotionisessential:
calledaweightsystemoflengthdandtotalweight|Q|:=i=0qi.Thereduction
Definition3.6.1.AfamilyofpositiverationalnumbersdQ:=(q0,...,qd)is
QredofQisdefinedastheuniqueprimitivelatticepointinpos(Q),andthe
factorλQ∈Q>0ofQisdefinedbyQ=λQQred.Twoweightsystemsare
regardedtobeisomorphic,iftheirentriesarejustpermutated.
LetQconsistonlyofnaturalnumbers.ThenwehaveλQ=gcd(q0,...,qd).
SuchaQiscalledreduced,ifλQ=1,andnormalized,ifafterremovingan
arbitraryweightwestillhaveareducedweightsystem.
Thereisnowthefollowingconnectiontolatticesimplices:
Definition3.6.2.LetP=conv(v0,...,vd)⊆MRbead-dimensionalrational
simplexwith0∈intP.

eReflexiv3.6.simplices

69

Thenwedefineqi:=|det(v0,...,vˆi,...,vn)|∈Q>0fori=0,...,nandthe
familyQP:=(q0,...,qd).QPiscalledtheassociatedweightsystemofPof
factorλP:=λQP.
Furthermoreweneedthefollowingdefinition:
Definition3.6.3.ForalatticepolytopeP⊆MRwegenerallydefinethe
sublatticeMPofMasthelatticegeneratedbytheverticesofP,i.e.,the
coarsestlatticesuchthatPisalatticepolytope.
Ad-dimensionallatticesimplexP⊆MRiscalledspanning,if0∈intPand
.M=MPWehavethefollowingcharacterization:
Lemma3.6.4.LetPbead-dimensionalsimplexwith0∈intP.Thenfor
QP=(q0,...,qn)wehave
dqivi=0.
=0iFurthermoreletPbealatticesimplex.Then(QP)red=QP/λP=(q0,...,qn)
istheuniquereducedweightsystemsatisyfing
dqivi=0.
=0iMoreoverλP=detMP=|M/MP|.EspeciallyPisaspanninglatticesimplex
ifandonlyiftheassociatedweightsystemisreduced.
Proof.Thisisthecontentof[Con02,Lemma2.4].Alsoobservethatthekernel
ofthesurjectivemapZd+1→MP=∼Zd,x→id=0xiviisfreeofrankone.
Whenconsideringsublatticesthefollowingdefinitionisveryconvenient:
Definition3.6.5(Hermitenormalform).Ford,λ∈N≥1wedenoteby
Herm(d,λ)thefinitesetoflowertriangularmatricesH∈Matd(N)withdeter-
minantλsatisfyinghi,j<hi,jforallj=1,...,d−1andi>j.
Theorem3.6.6.ForanyU∈Matd(Z)withdeterminantλ=0thereexistsa
matrixL∈GLd(Z)andaHermitenormalformmatrixH∈Herm(d,λ)such
.H=LUthatThisis[Con02,Thm.4.2].
ThereisnowthefollowingresultduetoConrads[Con02,3.6-3.8,4.4-4.7],see
also[Bat94,Thm.5.4.5].AsusuallywedenotebyP(Q)theweightedprojective
spaceassociatedtoaweightsystemQ∈Qd>0+1,e.g.,P(1,...,1)=Pd.
Conrads).yrev,(Bat3.6.7Theorem1.P→QPyieldsawell-definedcorrespondenceofisomorphismclassesof
spanninglatticesimplicesandreducedweightsystems.Wedenotethe
reversemapbyQ→PQ.

70

Chapter3.Reflexivepolytopes

2.HerebyPisaaFanopolytopeifandonlyifQPisanormalizedweight
system.FurthermoreP→X(ΣP,M)=∼P(QP)givesawell-definedcorrespon-
denceofisomorphismclassesofspanningFanosimplicesandisomorphism
classesofweightedprojectivespaces.
3.Anylatticesimplexcontainingtheorigininitsinteriorwithweightsystem
QistheimageofPQredunderaHermitenormalformmatrixofdeter-
minantλP.Inparticularthereareonlyfinitelymanylatticesimplices
containingtheorigininitsinteriorandhavingthesameassociatedweight
system.SuchalatticesimplexP⊆MRdefinesatoricvarietythatisthequotientof
theweightedprojectivespaceP(Q)bytheactionofthefinitegroupM/MP
oforderλP.
Howtoexplicitlyconstructtheuniquespanningsimplexassociatedtoa
reducedweightsystemisdescribedin[Con02].Notehoweverthatforgeneral
latticesimplicestheassociatedweightsystemdoesnothavetodeterminethe
uniquely!simplexlatticeThereisnowanimportantinvariantofaweightsystem,thisgeneralizes
[Con02,Def.5.4]aswillbeseenin3.6.26(1):
Definition3.6.8.LetQaweightsystemoflengthd.Thenwedefine
|Q|d−1
mQ:=q0∙∙∙qd∈Q>0.
Fortheproofofnextresultweneedalemmathatisprovenbyanexplicit
calculation:3.6.9.Lemman1−1−1∙∙∙∙∙∙−1
.........
det−1∙∙∙−1ni−1−1∙∙∙−1
.........
−1∙∙∙∙∙∙−1nd−1
d=n1∙∙∙nd−n1∙∙∙nˆj∙∙∙nd
=1jNowwecanaddsomeinterestingadditionalinformationthatismissingin
]:[Con02erpaptheProposition3.6.10.LetP⊆MRad-dimensionalsimplexwith0∈intP.
ThenQP∗=mQPQP.

simpliceseReflexiv3.6.

71

Proof.LetQ:=QP,t:=|Q|,V(P)={v0,...,vd}andfori=0,...,dwe
denotebyFi=conv(v0,...,vˆi,...,vd).FixabasisofManditsdualbasisofN.
Nowfixi∈{0,...,d},andletAibethematrixconsistingofthecoordinatesof
jv1,...,vˆi,...,vdasrows,andletBibethematrixconsistingqofthecoordinates
ofηF1,...,ηFˆi,...,ηFdascolumns.Sincevi=−j=iqivj,wegetηFi,vi=
j=iqi=qi=qi−1.
qjt−qit
Withoutrestrictionweseti=0.ApplyingthepreviouslemmatoA0B0,
wegetdet(A0B0)=q1∙∙∙qd−j=1q1∙∙∙qj∙∙∙qd=q1∙∙∙qd(t−j=1qj)=
ttdttˆttd−1d
d−1d−1
q1t∙∙∙qdq0.Thereforedet(B0)=detdet((AA00B)0)=q1t∙∙∙qd.

Inparticularweobservethatthereisaninvolution:
1−d(R\{0})d+1→(R\{0})d+1:x→(x0+∙∙∙+xd)x.
x0∙∙∙xd
Sothismotivatestodefineadualityalsoonthelevelofweightsystems:
Definition3.6.11.ForaweightsystemQwedefinethedualweightsystem
Q∗:=mQQformQasabove.Thisisadualityinthesense(Q∗)∗=Q.
IfQ=λQQredisaweightsystem,then
Q∗=mQredQred.(3.5)
λQThepropositioncannowbereformulatedasQP∗=(QP)∗.
Thisyieldsaconditionforself-duality(useabovetheorem):
Corollary3.6.12.LetPbealatticesimplexwith0∈intP,Q:=QP.
IfPisself-dual,i.e.,P=∼P∗,thenQ∗=Q,i.e.,λP=√mQred.
IfPisaspanninglatticesimplex,thereverseholds(withλP=1=mQred).

Inthesituationofareflexivesimplexthefollowingdefinitionturnsoutto
beconvenient:
Definition3.6.13.Aweightsystemiscalledreflexive,ifitisreducedandany
weightisadivisorofthetotalweight.Especiallyithastobenormalized.
Thenotionofareflexiveweightsystemismotivatedbythefollowingresult
[Con02,Prop.5.1](partially[Bat94,Thm.5.4.3]):
Theorem3.6.14(Batyrev,Conrads).UnderthecorrespondenceofTheorem
3.6.7wegetcorrespondencesofisomorphismclassesof
∙reflexivesimpliceswhoseverticesspanthelattice
systemsweighteflexiver∙∙weightedprojectivespaceswithGorensteinsingularities
Summingthisdiscussionupwegetageneralizationof[Con02,5.3,5.5]:

72

Chapter3.Reflexivepolytopes

Proposition3.6.15.LetP⊆MRbeareflexivesimplexwithassociatedweight
systemQ:=QP=λPQred.ThenQredisareflexiveweightsystem.
LetPred⊆MRbethereflexivesimplexcorrespondingtoQred.Then:
1.(QP∗)red=Qred,λP∗=mQPλP=mλQPred,mQred=λ(Pred)∗∈N>0.
2.mQredPredλ→PPλ→P(Pred)∗.
forinjectivelatticehomomorphismsofthegivenintegerdeterminant.
3.λP|mQred.Furthermore
λP=1⇐⇒P∼=Pred,λP=mQred⇐⇒P=∼(Pred)∗.
Proof.From[Con02,5.1]wegetthatQredisareflexiveweightsystem(aneasy
calculation).1.Followsfrom3.6.10and3.6.4.2.Firstapply3.6.7toP.Thenapply
3.6.7tothelatticesimplexP∗,use1.anddualize.3.from2.
From(3.3)and3.6.7(3)wegetthefollowingcorollary:
Corollary3.6.16.GorensteintoricFanovarietieswithclassnumberoneare
uniquelyassociatedtofansspannedbythefacesofareflexivesimplex.
AnysuchvarietyisthequotientofaweightedprojectivespaceP(Q)with
Gorensteinsingularitiesbytheactionofafiniteabeliangroupoforderlessor
equaltomQ.Equalityholdsiffthevarietyisassociatedtothefanspannedby
thefacesof(PQ)∗.
Notehowever,thatifaspanninglatticesimplexPhasa(reduced)weight
systemQsuchthatQ∗isalsoaweightsystemwithonlynaturalnumbers,i.e.,
mQ∈N,thenPdoesnotnecessarilyhastobereflexive,i.e.,Qdoesnothave
tobereflexive,e.g.,Q=(1,1,1,9)withmQ=16andQ∗=(16,16,16,144).
Nextweareconcernedwiththequestionhowtoconstructlatticesimplices
fromweightsystems.Thisisactuallyanon-trivialtask,howeveratfirstglance
thereseemstobeanaivewayofdoingit:
Definition3.6.17.LetQ=(q0,...,qd)beareducedweightsystem.Define
(k0,...,kd):=(|Q|/q0,...,|Q|/qd)∈Qd>0+1.Weassumekd=max(k0,...,kd).
WedefineCQ:=conv(conv(0,k0e0,...,kd−1ed−1)∩M)⊆MR,wheree0,
...,ed−1isanarbitrarybutfixedZ-basisofM.Fore:=e0+∙∙∙+ed−1we
denotebySQ:=CQ−ean(uptolatticeisomorphismwell-defined)lattice
polytopeassociatedtotheweightsystemQ.
NotethatSQmaynotbeasimplexanymore(e.g.,Q=(6,2,1))!However:
Proposition3.6.18.LetQbeareflexiveweightsystemwithminimalweight
qd.Thenwehaveinthenotationofthepreviousdefinitionthat
SQ=∼conv(k0e0−e,...,kd−1ed−1−e,−e)
isalatticesimplexwithassociatedweightsystemqdmQQ.

simpliceseReflexiv3.6.

eurthermorFSQisreflexive⇐⇒qd=1⇐⇒SQ=∼(PQ)∗.
asecthisInPQ=∼conv(e0,...,ed−1,−q0e0−∙∙∙−qd−1ed−1).

73

Proof.ObviouslyS:=SQ=conv(k0e0−e,...,kd−1ed−1−e,−e)isalattice
simplexsatisfying1/k0(k0e0−e)+∙∙∙+1/kd−1(kd−1ed−1−e)+1/kd(−e)=0.
UsingLemma3.6.4wegetQS=λSQ.UsingLemma3.6.9wegetλSq0=
|det(k1e1−e,...,kd−1ed−1−e,−e)|=k1∙∙∙kd−1,henceλS=k0∙∙∙|Qk|d−1=
|Q|d−1=qdmQ.
q0∙∙∙Fqdor−1thesecondpartwecanassumebythepreviouspropositionthatqd=1,
andwehavetoshowthatSisreflexive.Nowwesimplyobservethat−q0e0−
∙∙∙−qd−1ed−1,e0,...,ed−1aretheinnernormalsofS.
Soweseethatunderthisnaiveconstructionreflexiveweightsystemsnot
necessarilyyieldreflexivesimplices,e.g.,lookatQ=(4,3,3,2);onlyinthe
(moreorlesstrivial)casewheretheverticesofafacetofthecorresponding
spanningreflexivesimplexformalatticebasis.Soweget:
Corollary3.6.19.Q→SQ∼=(PQ)∗yieldsacorrespondenceofreflexiveweight
systemscontaining1asanentryandthedualsofreflexivesimplicescontaining
afacetwhoseverticesformalatticebasis.
resultmainThe3.6.2Weneedthefollowingwell-knownsequence(e.g.,see[AS70]):
Definition3.6.20.Therecursivesequence[Slo04,A000058]ofpairwiseco-
primenaturalnumbersy0:=2,yn:=1+y0∙∙∙yn−1iscalledSylvestersequence.
Itsatisfiesyn=yn2−1−yn−1+1andstartsasy0=2,y1=3,y2=7,y3=43,
y4=1807.Wealsodefinetn:=yn−1=y0∙∙∙yn−1.
Usingthesenumberswedefinetwospecialsetsofreflexiveweightsystems:
3.6.21.Definition∙Thed+1-tupleofnaturalnumbers
Qd:=(td,...,td,1)
y0yd−1
iscalledSylvesterweightsystemoflengthd.
∙Thed+1-tupleofnaturalnumbers
Qd:=(2td−1,...,2td−1,1,1)
y0yd−2
iscalledenlargedSylvesterweightsystemoflengthd.

74

Chapter3.Reflexivepolytopes

Thegoalofthissectionistoprovethefollowingtheorem:
3.6.22.Theorem1.IfQisareflexiveweightsystemoflengthd,then
|Q|≤td,withequalityiffQisisomorphictoQd.
2.IfPisareflexivesimplexford≥3,then
|QP|≤2td2−1,
withequalityiffP=∼SQredforQredisomorphictoQdor(3,1,1,1).
3.IfPisareflexivesimplex,then
|QP||QP∗|≤td2,
withequalityiffP=∼SQd(=∼SQ∗d).
SincebyTheorem3.6.7(3)(seealso3.6.15and3.6.16)thereareonlyfinitely
manyreflexivepolytopeshavingasthereductionoftheirassociatedweight
systemthesamereflexiveweightsystemwegetfromthefirstpointadirect
proofofafactthatisknowntoholdingeneralforreflexivepolytopes:
Corollary3.6.23.Thereisonlyafinitenumberofisomorphismclassesof
d-dimensionalreflexivesimplices.
Inparticularthereisaclassificationalgorithmasdescribedin[Con02]:First
determineallreflexiveweightsystemsQoflengthd.Thenconstructtheasso-
ciatedspanningreflexivesimplexPQ.Eventuallywelookforreflexivesimplices
intheimagesofPQunderanyHermitenormalformmatrixofdeterminantλ
forλadivisorofmQ,whereλ≤mQ/2sufficesbyduality.
Nowtheproofofthetheoremisessentiallypurenumbertheory.
Forthisweneedthenotionofaunitpartitionthatiscloselyrelatedtothat
ofanEgyptianfraction,e.g.,see[Epp04]:
Definition3.6.24.Afamilyofpositivenaturalnumbers(k0,...,kd)iscalled
aunitpartitionoftotalweightlcm(k0,...,kd),ifid=01/ki=1.
Thecrucialobservationisthefollowingresult(essentiallyduetoBatyrevin
5.4]):,[Bat94Proposition3.6.25.Thereisaabijectionbetweenreflexiveweightsystems
andunitpartitions,givenbymappingQ∈Qd>0+1to(|qQ0|,...,q|Qd|),respectively
mapping(k0,...,kd)to(lcm(kk00,...,kd),...,lcm(kk0d,...,kd)).
Herebythelengthandthetotalweightofthereflexiveweightsystemandthe
correspondingunitpartitionarethesame.
FromPropositions3.6.15and3.6.12wegetthefollowingcorollary:
Corollary3.6.26.LetP⊆MRbeareflexivesimplexwithQ:=QP.Let
Pred⊆MRbethereflexivesimplexcorrespondingtoQredand(k0,...,kd)the
associatedunitpartition.Then

simpliceseReflexiv3.6.

75

1.k0∙∙∙kd
mQred=lcm(k0,...,kd)2=λ(Pred)∗∈N>0.
∗2.IfPself-dualisifself-dualandonly(i.e.,ifPk∙=∼∙∙Pk),=thenlcm(kk0∙,∙.∙.k.,dkhas)2.tobeasquare.Predis
d0d0Inparticularthefirstpointyieldsthepurelynumber-theoreticcorollarythat
theThefirstsquarepartofofthethisleastcorollarycommonwasdivisoralreadyofaprounitvenbypartitionBatyrevdividesinthe[Bat94pro,duct.Cor.
5.5.4]wheninvestigatingfundamentalgroups.
Nowwedefinetheunitpartitionscorrespondingtoaboveweightssystems:
3.6.27.Definition∙(y0,...,yd−1,td)iscalledSylvesterpartitionoflengthd(correspondingto
itQd).correspItisaondsunittoapartitionself-dualoftotalreflexivweeightsimplextd=Std−∼1yPd−.1.SincemQd=1,
=QQ∙(y0,...,yd−2,2td−1,2td−1)iscalledenlargedSylvesterpartitionoflength
d(correspondingtoQd).NotethatmQd=td−1.
MelnikTheseovintwo[HM04unit](inpartitionsturnwbasingereondefined[LZ91],upon[Hen83observ],ations[PWZ82b]).yHaaseand
Asanillustrationofthepreviousnotionsweclassifythefivetwo-dimensional
simplices:ereflexivExample3.6.28.Considerthecased=2.Wehavethreeunitpartitions:
1.(3,3,3)correspondingtoQ:=(1,1,1).ThisyieldsmQ2=3.Sowe
(haPQve)∗P=∼Q:=convcon((2v,−((1,1),0)(,−(01,,1)2),,((−−11,,−−1))1))as(corresptheonlyondingreflexivtoPe)andsimplicesSQ=P
with(QP)red=Q(dueto3.6.15(3)):

(index 3)

39=3*2.TheSylvesterpartition(2,3,6)correspondingtoQ:=Q2=(3,2,1).∼This
yieldsmQ=1.Sowegettheself-dualreflexivesimplexPQ=SQ=
conv((1,0),(0,1),(−3,−2))(correspondingtoP(Q)):

6d=(6d)*

(index 1)

76

Chapter3.Reflexivepolytopes

3.TheenlargedSylvesterpartition(2,4,4)correspondingtoQ:=Q2=
(2,1(corresp,1).ThisondingtoyieldsP(mQQ))=and2.SSoQw=e(PhaQv)e∗P=∼Q:=convcon((1v,−((1,1),0)(,−(01,,1)3),,((−−12,,−−1))1))
astheonlyreflexivesimplicesPwith(QP)red=Q:

4c

8c=(4c)*

(index 2)

UsingProposition3.6.15,Corollary3.6.26andEquation(3.5)onp.71itis
ofstraighthefollotforwwingardntoumbdeduceer-theoreticTheoremprop3.6.22osition:fromthesecondandthirdstatements
Proposition3.6.29.Let(k0,...,kd)beaunitpartition.
1.d+1≤max(k0,...,kd)≤td,
withequalityinthesecondcaseonlyfortheSylvesterpartition.
Furthermore:Ifk0≤...≤kd,then
kj≤(d−j+1)tj
forj∈{0,...,d}.
2.(d+1)2≤lcm(k0,...,kd)2≤k0∙∙∙kd≤td2,
withequalityinthelastcaseonlyfortheSylvesterpartition.
3.Letd≥3andk0≤...≤kd.Then
lcm(kk00∙,∙.∙.k.d,kd)≤k0∙∙∙kd−1≤2td2−1,
itywheriffe(thek0,.first..,ekd)qualityistheholdsenlariffgekdd=lcmSylvester(k0,p...,artitionkd),orand(2,the6,6se,c6)ond.equal-
Fortheproofweneedseveralresultsaboutsumsofunitfractions.
proofTheismostratherimportancomplicated.tonewasgivenbyCurtiss[Cur22,Thm.I],howeverhis
Theoremn13.6.30(Curtiss).Letx1,...,xnbepositiveintegerssuchthat
s:=i=1xi<1.Thenn−1
11s≤i=0yi=1−tn,
withequalityifandonlyif{x1,...,xn}={y0,...,yn−1}.

simpliceseReflexiv3.6.

77

Inthecaseofaunitpartitionthereisalsoaveryinsightfulandeasyproof
thatisstraightforwardtodeducefromthefollowingniceresult[IK95,Lemma
1].Herewehaveincludedstatementsthatareimplicitintheirproof.
Lemma3.6.31(Izhboldin,Kurliandchik).Letx1,...,xnberealnumbers
satisfyingx1≥x2≥∙∙∙≥xn≥0,x1+∙∙∙+xn=1andx1∙∙∙xk≤xk+1+
∙∙∙+xnfork=1,...,n−1.Then
11xn≥tn−1,x1∙∙∙xn≥t2,
1−nwhereequalityinthefirstcaseholdsiffequalityinthesecondcaseholdsiff
xi=yi1−1fori=1,...,n−1.
Wecannowmakeanadditiontothisresult.Forthisweneedthefollowing
y:inequalitLemma3.6.32.Letn≥4,1≤k≤n−1.Then
(k+1)ktnk−+1k−1≤2tn2−2,
withequalityiffk=1or(n,k)=(4,2).
Proof.Proofbyinductiononn.Byexplicitlycheckingn=4,5,wecanassume
6.n≥Fork=1thestatementistrivial,soletk≥2.Byinductionhypothesis
for(n−1,k−1)wehavekk−1tnk−k−1≤2tn2−3,thisyields(k+1)ktnk−+1k−1≤
2tn2−3tn−k−1(kk+1)kk.Since(kk+1)k<e,itisenoughtoshowtn2−3tn−k−1ek≤
tn2−2,orequivalently,tn−k−1ek≤yn2−3.
Forn≥6itiseasytoseethate(n−1)≤yn−3(e.g.,by3.6.33below).
Hencetn−k−1ek≤tn−3e(n−1)≤tn−3yn−3<yn2−3.
ThefollowinglemmagivestheasymptoticalbehavioroftheSylvesterse-
(4.17)]):,[GKP89(e.g.,quenceLemma3.6.33.Thereisaconstantc≈1.2640847353∙∙∙(calledVardicon-
stant,see[Slo04,A076393])suchthatforanyn∈N
yn=c2n+1+1.
2

Using3.6.32andtheideasoftheproofof3.6.31wecannowshow:
Lemma3.6.34.Letn≥4,x:=(x1,...,xn)asinLemma3.6.31.Then
1x1∙∙∙xn−1≥2t2,
2−nwithequalityiff(1/x1,...,1/xn)equals(2,6,6,6)ortheenlargedSylvesterpar-
titionoflengthn−1.

Chapter3.Reflexivepolytopes

78Chapter3.Reflexivepolytopes
of.oPrI:STEPLetAdenotethesetofn-tupelsxsatisfyingtheconditionsofthelemma.
Itiseasytoseethatwehaveforx∈Anecessarily1>x1andxn>0.
SinceAiscompact,thereexistssomex∈Awithx1∙∙∙xn−1minimal.
11111Becauseof(y0,...,yn−3,2tn−2,2tn−2)∈Awehavex1∙∙∙xn−1≤2t2.
2−nLetusassumethatxn−1>xn.
Claim:Inthiscasewehave
x1>x2>∙∙∙>xn−1>xn,
x1∙∙∙xk=xk+1+∙∙∙+xnfork=1,...,n−2.
claim:ofofoPrByconventionweletx0:=1andxn+1:=0.Weproceedbyinductionon
l=1,...,n−1,andassumebyinductionhypothesisthatx1>x2>∙∙∙>xl≥
xl+1andx1∙∙∙xk=xk+1+∙∙∙+xnfork=1,...,l−2.
cases:threedistinguisheW1.xl>xl+1andx1∙∙∙xl−1=xl+∙∙∙+xn.
Inthiscasewecanproceed.
2.xl>xl+1andx1∙∙∙xl−1<xl+∙∙∙+xn.
Thisimpliesl≥2.Thenwecanfindsomeδ>0s.t.x∈Awith
xl−1:=xl−1+δ,xl:=xl−δandxj:=xjforj∈{1,...,n}\{l−1,l}.
x∙∙∙xHencex1∙∙∙xn−1=1xl−1nxl−1(xl−1xl+δ(xl−xl−1)−δ2)<x1∙∙∙xn−1,a
tradiction.con3.xl=xl+1=∙∙∙=xi>xi+1forl+1≤i≤n−1.
Thisimpliesl≤n−2.Againwefindsomeδ>0s.t.x∈Awith
xl:=xl+δ,xi:=xi−δandxj:=xjforj∈{1,...,n}\{l,i}.
Thiscanbedone,sinceotherwisetherehastoexistl≤j<isuch
thatx1∙∙∙xj=xj+1+xj+2+∙∙∙+xn.Sincexj=xj+1wehave0=
(1−x1∙∙∙xj−1)xj+xj+2+∙∙∙+xn,acontradiction.
Sinceagainx1∙∙∙xn−1<x1∙∙∙xn−1,wegetacontradiction.
Endofproofofclaim.
Sowehavex1=x2+∙∙∙+xn=1−x1,hencex1=21=y10.Byinduction
onk=2,...,n−2wegetx1∙∙∙xk=xk+1+∙∙∙+xn=1−x1−∙∙∙−xk,hence
1111tk−1xk=1−y0−∙∙∙−yk−2−xk=tk−1−xk,soxk=1−tk−1xk.Thisimplies
1111xk=1+tk−1=yk−1.Sothisyieldsx1=y0,...,xn−2=yn−3.
Furthermorexn−1+xn=1−x1−∙∙∙−xn−2=tn1−2.Sincexn−1>xn,weget
11xn−1>2tn−2.Thereforewehaveprovenx1∙∙∙xn−1>2tn2−2,acontradiction.
Sothisstepyieldsxn−1=xn.

simpliceseReflexiv3.6.

79

I:ISTEPLetAdenotethesetof(n−1)-tupelsw∈Rn−1satisfyingthefollowing
conditions:w1≥w2≥∙∙∙≥wn−2≥wn−1≥0,w1+∙∙∙+wn−1=1and
2w1∙∙∙wk≤wk+1+∙∙∙+wn−1fork=1,...,n−2.
Nowletw∈Abefixedwithw1∙∙∙wn−1minimal.
1111Since(y0,...,yn−3,tn−2)∈A,wehavew1∙∙∙wn−1≤t2.
2−nLetz:=ws=∙∙∙=wn−2=wn−1for1≤s≤n−1minimal.
2Wedefinek:=n−s.Therearethreecasestoconsider:
1.s=1,i.e.,k=n−1.
Thennz=(n−2)z+2z=w1+∙∙∙+wn−2+wn−1=1,soz=1/n.This
212impliesw1∙∙∙wn−1=nn−1.Howevert2<nn−1forn≥4by3.6.32,a
2−ntradiction.con2.s=2,i.e.,k=n−2.
Then1=w1+w2+∙∙∙+wn−2+wn−1=w1+(n−3)z+2z=w1+(n−1)z,
1hencew1=1−(n−1)z.Sincew1>z,wegetz<n.Ontheotherhand
w1≤w2+∙∙∙+wn−1=(n−1)z,hencez≥2(n1−1).
Wehavew1∙∙∙wn−1=(1−(n−1)z)2zn−2.Thisfunctionattainsits
111minimumontheinterval[2(n−1),n[onlyforz=2(n−1).(Theproofof
thisstatementislefttothereader.)
11Thereforetn2−2≥w1∙∙∙wn−1≥(2(n−1))n−2.Howeverby3.6.32wehave
11(2(n−1))n−2≥tn2−2,withequalityonlyforn=4.Hencewegetn=4,
z=2(n1−1),andw=(21,61,31).
3.s≥3,i.e.,k≤n−3.
Nowasimilarreasoningasintheproofofaboveclaimyields
11w1=,...,ws−2=.
y0ys−3
1Then1=w1+∙∙∙+ws−2+ws−1+ws+∙∙∙+wn−1=1−ts−2+ws−1+
1(n−s+1)z,hencews−1=ts−2−(k+1)z.
1Sincews−1>z,wegetz<(k+2)ts−2.
Sincew1∙∙∙ws−2ws−1≤ws+∙∙∙+wn−1,wegetts1−2(ts1−2−(k+1)z)≤
11(k+1)z,hencez≥(k+1)ts2−2(1+t1)=(k+1)ts−1.
2−ss−n11Nowwehavew1∙∙∙wn−1=ts−2(ts−2−(k+1)z)2z=:f(z).
Sincethefunctionf(z)isforz>0strictlymonotoneincreasingupto
somevalueandthenstrictlymonotonedecreasing,weseethat
111tn2−2≥w1∙∙∙wn−1≥min(f((k+1)ts−1),f((k+2)ts−2))=
min(2,2).
(k+1)ktsk−+11(k+2)k+1tsk−+22

80

Chapter3.Reflexivepolytopes

Therearetwocases:
(a)s=n−1,i.e.,k=1.
2121Herew1∙∙∙wn−1≥min(tn2−2,9tn3−3).Sincetn2−2≤9tn3−3(bythe
11recursivedefinition),wegetw1∙∙∙wn−1=t2,andz=2tn−2.
111n−2
Hencew=(y0,...,yn−3,tn−2).
(b)s≤n−2,i.e.,k≥2.
22ysk−+12(k+2)k+1
Here(k+1)ktk+1≤(k+2)k+1tk+2ifandonlyifts−2≥(k+1)k.This
s−1s−2
istruefork=2.Fork≥3wehave(k+2)(k+2)k<(k+2)e≤3k≤
+1kkysk−+1212
ys−2<ts−2.Hencet2≥w1∙∙∙wn−1≥(k+1)ktk+1,acontradiction
n−2s−1
3.6.32.toI:IISTEPNowwecanfinishtheproof:Sincew:=(x1,...,xn−2,2xn−1)∈A,weget
wn−11
x1∙∙∙xn−1=w1∙∙∙wn−2≥2,
22tn−2
1111whereequalityimpliesthat(x1,...,xn)=(y0,...,yn−3,2tn−2,2tn−2)orn=4
and(x1,x2,x3,x4)=(1/2,1/6,1/6,1/6).

Itisnowstraightforwardtoprovethemainresultofthissection:

ProofofProposition3.6.29.Wecanassumek0≤...≤kd.Firstweshowthat
forxi:=1/kitheconditionsin3.6.31aresatisfied:PSoleti∈{0,...,d−1}.
Then0<xi+1+∙∙∙+xd=1−x0−∙∙∙−xi=k0∙∙∙ki−kji∙∙∙=0kk0∙∙∙kˆj∙∙∙kd≥k∙∙∙1k=
x0∙∙∙xi.0i0i
1.Thefirstlowerboundcanbeimmediatelyderivedfromthepartition
property.Theupperboundsareprovenbyshowing
111kj≥d−j+1tj
forj∈{0,...,d}.
Forj=0theresultistrivialsincet0=1,1/k0+∙∙∙1/kd=1andk0=
i=01/ki≤1−1/tj,so1/kj+∙∙∙+1/kd≥1/tj,hencetheorderingassumption
minj−(1k0,...,kd).Forj∈{1,...,d}Lemma3.6.30implies1−1/kj−∙∙∙−1/kd=
t.statemendesiredtheyields2.Thelowerboundfollowsfrom1.,themiddleboundfrom3.6.26(1).The
upperboundfollowsfrom3.6.31.
3.Thisfollowsimmediatelyfrom3.6.34.

3.7.Latticepointsinreflexivepolytopes

81

3.7Latticepointsinreflexivepolytopes
Itisinterestingtotrytofindsharpupperboundsonthenumberoflattice
pointsinad-dimensionalreflexivepolytope.Ontheonehandthisismotivated
byconvexgeometry(seeforinstance[LZ91])andgeometryofnumbers(see
theresultsalgebraic-geometricintheinpreviousterpretation:section),LetonPthe⊆Motherbehandreflexivtheree,thenisbalsoya(1.7)direct
R|P∩M|=h0(XP,−KXP).(3.6)
boundingAnothertheimpvolumeortantofinavarianlatticetofpaolytoplatticeealsopbolytopoundseistheitsvnumolume;berofmoreolatticever
points(e.g.,[LZ91]):
Lemma3.7.1(Blichfeldt).LetP⊆MRbead-dimensionallatticepolytope.
Then|P∩M|≤d+d!vol(P).

IfP⊆MRisreflexive,thenthenormalizedvolumeofPisjustthe(anti-
canonical)degreeofXP(seeProp.2.3.15):
deg(XP)=(−KXP)d=d!vol(P).(3.7)
degreeInofalgebraicaGorensteingeometryFanooneisvinarietyterestedwithincanonicalfindingasharpsingularitiesupperb(seeound[Pro04onthe]).
Theresultsachievedherecanbeusedasaconjectureforthismoregeneral
situation.topesIntheeffectsfirstthesubsectionEhrhartpweshoolynomial.whowInthethedualitsecondypropweertproyvofeuppreflexiverbepoundsoly-
onandthevdetermineolumeandthehencemaximalonntheumnberumboferoflatticelatticepoinptsoinontsofanaedge.reflexivInethesimplexlast
Thissubsectionisusefulweindescribtheecasehoofwcentotrallycountlatticesymmetricpoinortsmoterminalduloareflexivnaturalepnolytopumbes.er.
olynomialpEhrhartThe3.7.1Insectionthis1.6.subsectionThewtheell-knotopicwnoffactislatticepresenpointed,tsinhowptheolytopessymmetryiscontinpropuedertyfromof
1.5.3).reflexiveInpthreeolytopescandimensionsbeafoundsimpleagainandinalsothewell-knoEhrhartwnpPickolynomialtypeform(seeulaThm.for
thevolumeofareflexivepolytopeisderived.Thisresultiscertainlyfolklore,
howevertheauthorcouldnotfindanexplicitreference.
Thereisthefollowingwell-knownresult(e.g.,see[Has00],[HM04])with
mostpartsoriginallyduetoHibi[Hib92]:
intPropP.Theositionfollowing3.7.2.cLetonditionsP⊆arMeRebedquivalent:-dimensionallatticepolytopewith0∈
eflexiverisP1.2.eP(k)=|relint((k+1)P)|forallk∈N

82

Chapter3.Reflexivepolytopes

3.eP(k)=(−1)deP(−k−1)forallk∈N
4.rvol(P)=d1F∈F(P)rvol(F)
5.coeffd−1(eP)=2dcoeffd(eP)
Fortheprooffirstobservethatanyfull-dimensionalconeinMRcontainsa
latticebasis.Fromthisweeasilydeduce:
Lemma3.7.3.Pasintheproposition,F∈F(P),νF∈NQwithνF,F=1.
Thefollowingconditionsareequivalent:
Nν1.F∈2.νF,m∈Zforallm∈pos(F)∩M
3.pos(F)∩M⊆∪k∈NkF
UsingthelemmaandthereciprocitylawinTheorem1.5.3thefirstthree
equivalencesarestraightforwardtoprove.Theremainingequivalencesareeasy
toseeby3.1.4andtheresultsinsection1.5.
Inparticularwegetbythethirdequivalenceintheproposition:
Corollary3.7.4.TheEhrhartpolynomialofad-dimensionalreflexivepolytope
isdeterminedbyitsvaluesfork=1,...,2d.
Ford=2,3thecorollaryyieldsthatePisdeterminedbyeP(1)=|P∩M|:
Corollary3.7.5.LetP⊆MRbead-dimensionalreflexivepolytope.
Ifd=2,then
eP(x)=(|P∩M|−1)x2+(|P∩M|−1)x+1,
2222Inparticularvol(P)=(|P∩M|−1)/2.
Ifd=3,then
eP(x)=(|P∩M|−1)x3+(|P∩M|−3)x2+(|P∩M|+3)x+1,
26223Inparticularvol(P)=|P∩M|/3−1.
Proof.JustchecktheequalityofbothsidesusingProposition3.7.2.

Inhigherdimensionsthereisnosuchdirectrelationbetweenthevolumeand
thenumberoflatticepointsofareflexivepolytope(e.g.,seepage88).
Analternativeproofofthisformulainthethree-dimensionalcasecouldhave
beengivenbyusingthetwo-dimensionalformulaofPickforlatticepolygons
andthefactthatthefacetshavelatticedistanceonefromtheorigin.Apurely
algebraic-geometricproofwouldbepossiblebyprovingtheformulafirstfora
smoothFanopolytopeusingRiemann-Rochandthedouble-weight-formulaof
Oda[Oda88,Cor.1.32]andthenusingtheexistenceofacrepantresolutionin
3.1.6.Applyingthesocalled”ArithmeticEuler-Poincare´formula”asgivenin
[Kan98,Theorem6]byKantorweget(recallDefinition3.1.7):

3.7.Latticepointsinreflexivepolytopes

83

Corollary3.7.6.Inanytriangulationoftheboundaryofathree-dimensional
reflexivepolytopeP⊆MRofVolumeVsuchthatanytwo-dimensionalsimplex
isanemptylatticepolygonthenumbersfi(i=0,1,2)ofi-dimensionalsimplices
satisfy:f0=|P∩M|−1=3V+2,f1=9V,f2=6V.

3.7.2Boundsonthevolumeandlatticepoints
Throughoutthesectionletd≥2.
Inthissubsectionwewillprovethreeresultsthatwillbefirstformulatedin
algebraic-geometriclanguageandthenintheconvex-geometricsetting.They
giveupperboundson
1.theanticanonicaldegreeofaGorensteintoricFanovarietyXofclass
numberone(Thm.A),respectivelythevolumeofareflexivesimplexP
A’);(Thm.2.theanticanonicaldegreeofatorus-invariantcurveConX(Thm.B),
respectivelythenumberoflatticepointsonanedgeofareflexivesimplex
B’);(Thm.P3.theproductofthedegreesofthedualpairXandX∗(Thm.C),respec-
tivelytheproductofthevolumesofPandP∗(Thm.C’).
Nowwegivethefirstalgebraic-geometricresult,hereasinthewholesub-
section,weusethenotationfromtheprevioussection:
Theorem3.7.7(A).LetXbead-dimensionalGorensteintoricFanovariety
one.ernumbclasswith1.Ifd=2,then
(−KX)2≤9,
withequalityiffX=∼P2.
2.Ifd=3,then
(−KX)3≤72,
withequalityiffX=∼P(3,1,1,1)orX=∼P(Q3)=P(6,4,1,1).
3.Ifd≥4,then
(−KX)d≤2td2−1,
withequalityiffX=∼P(Qd).
Thismotivatesthefollowingconjecture:
Conjecture3.7.8.TheresultsoftheoremAholdforGorensteinFanovarieties
singularities.canonicalwith

84

Chapter3.Reflexivepolytopes

InthecaseofthreefoldstheboundintheoremAistheso-calledFano-
Iskovskikhconjecture.IthasveryrecentlybeenprovenbyProkhorov[Pro04].
DuetoasharpversionoftheConeTheorem(p.23)forprojectivetoric
varietiesasgivenin[Fuj03]weknowthatthereisalwaysatorus-invariant
integralcurveonXsuchthatitsanticanonicaldegreeisatmostd+1.The
nexttheoremshowsthatthereisisageneralupperboundinoursetting:
Theorem3.7.9(B).LetXbead-dimensionalGorensteintoricFanovariety
withclassnumberone.LetCbeatorus-invariantintegralcurveonX.Then
−KX.C≤2td−1,
whereequalityimpliesX=∼P(Qd).
LetXbead-dimensionalGorensteintoricFanovarietywithclassnumber
one.ThenX=XPforsomereflexivesimplexP⊆MR(seeCor.3.6.16).
Definition3.7.10.WedefineX∗astheGorensteintoricFanovarietywith
classnumberonethatisassociatedtothefanspannedbythefacesofP.
Thereisalsothefollowingresult:
Theorem3.7.11(C).LetXbead-dimensionalGorensteintoricFanovariety
withclassnumberone.Then
(−KX)d(−KX∗)d≤td2,
withequalityiffX=∼P(Qd).InthiscaseX=∼X∗.
FurthermoreletXbeaweightedprojectivespacewithGorensteinsingulari-
Thenties.(−KX∗)d≤td,
withequalityiffX=∼P(Qd).
OnecouldconjecturethattheoremBandthefirstpartoftheoremCmight
alsobetrueforGorensteintoricFanovarietieswitharbitraryclassnumber.
Nowthenextobservationshowshowtoapplytheresultsoftheprevious
section:Lemma3.7.12.LetP⊆MRbealatticesimplexwithassociatedweightsystem
QP=(q0,...,qd).Then
dvol(P)=qd=|QP|.
i=0d!d!
FromProp.3.6.15(2)weseethatifQistheassociatedreflexiveweight
systemofP(i.e.,QisthereductionofQP),thenvol(P)≤vol((PQ)∗).Moreover
ifmin(Q)=1,∼asfor∗QdorQd,thenrecallfromDefinition3.6.17andProp.
3.6.18thatSQ=(PQ).
By(3.7)onp.81theoremAcanbederivedfromthefollowingconvex
result:geometric

3.7.Latticepointsinreflexivepolytopes

85

(A’).3.7.13Theorem1.S(1,1,1)istheuniquetwo-dimensionalreflexivesimplexwiththelargest
volume29,respectivelythelargestnumberoflatticepoints10.
2.S(3,1,1,1)andSQ3aretheonlythree-dimensionalreflexivesimpliceswith
thelargestvolume12,respectivelythelargestnumberoflatticepoints39.
3.Letd≥4.ThenSQdistheuniqued-dimensionalreflexivesimplexwiththe
largestvolume2of2td2−1/d!.Anyd-dimensionalreflexivesimplexcontains
atmostd+2td−1latticepoints.
ProofoftheoremA’.ThisisstraightforwardfromExample3.6.28,Corollary
3.7.5,Theorem3.6.22(2),Lemma3.7.1andthepreviouslemma.

Theseboundsvastlyimproveonmoregeneralboundsonlatticesimplices
containingonlyonelatticepointintheinteriorasgiven(see[Pik00]).Herewe
citethefollowingtheoremfrom[LZ91]:
umeTheoremofacanonic3.7.14alFano(Hensleyp,olytopeinLagarias,MR.ThenZiegler).VisLetfiniteVbewiththemaximalvol-
V≤14d2d+1.
polytopAnyectoaanoniccalanonicFalanoFpanoolytoppeolytopcanebceembontaineedded)din(i.e.,theislatticeisomorphiccubeofassidealatticlengthe
atmostd∙d!V(respectivelyd!V,ifthepolytopeisasimplex).
Fromthistheoremwegetthefinitenessofisomorphismclassesofcanonical
polytopesinafixeddimension.
Theboundinthetheoremisextremelytoolargeatd+1leastforreflexivepoly-
1622topescanonicalinloFwanopdimensions,olygonhase.g.,atformostda=v2olumewegetof4.145,achiev=ed14by,thehowpeverolytopaney
oftype9inProp.3.4.1.
aInreflexivegeneralsimplexTheoremthanA’thegivespreviousalwaysatheorem!betterupperboundonthevolumeof
weseeUsingthatalowthereerbisoundstillaonsmallthengapumbtoerofbridge:latticepointsinSQddueto[PWZ82]
Corollary3.7.15.LetJdenotethemaximalnumberoflatticepointssome
d-dimensionalreflexivesimplexcanhave.Thenwegetford≥3
3(d1−2)!td2−1<J≤d+2td2−1∈O(c2d+1),
wherec≈1.26408istheVardiconstant(see3.6.33).
ThecomputerclassificationofKreuzerandSkarke[KS04b]yieldsthatthe
d=maximal2,39nforumbder=of3andlattice680poinfortsdin=a4.dThe-dimensionalcasesofreflexivequaliteypareolytoppreeiscisely10forthe
reflexivesimplicesgivenintheoremA’!Thismotivatesthefollowingconjecture:
dimensionalConjecturereflexiv3.7.16.epLetolytopde≥4:withThethelargestreflexivenumbsimplexerofSQdlatticeisptheoints.uniqued-

86

Chapter3.Reflexivepolytopes

Asacorollaryoftheprevioustwotheoremswecanlookatanembedding
intoamultipleoftheunitlatticecube[−1,1]d(seealsosection6.4):
Corollary3.7.17.Anyd-dimensionalreflexivesimplex(ford≥4)canbe
embeddedin[−l,l]dforsomel∈Nwithl≤2td2−1≤2c2d+1forc≈1.26408.
TheoremCtranslatesby(3.7)andTheorem3.6.14into:
Theorem3.7.18(C’).LetPbead-dimensionalreflexivesimplex.Then
vol(P)vol(P∗)≤td22,
!)d(∗∼∼withFequalityurthermoriffePlet=theSQd.verticHereseofSQPd=(generSQdate).thelatticeM.Then
vol(P)≤dtd!,
withequalityiffP=∼SQd.
ProofoftheoremC’.FollowsimmediatelyfromTheorem3.6.22(3)andLemma
3.7.12.

HerethefirstinequalitycanbeseenasageneralizationoftheBlaschke-
Santal´oinequality(1.6)insection1.5.
FinallyconcerningtheoremBweremarkthatanytorus-invariantintegral
curveCisgivenbyawallρ∈(d−2),i.e.,a(d−2)-dimensionalconeof.
MoreoverρisobviouslyincorrespondencewithanedgeeofP.Thereisthe
followingobservation(see[Lat96,Cor.3.6]):
−KX.C=|e∩M|−1.(3.8)
HereUsingtherightequationsideis(3.8)calledweseethethatlatticetheoremlengthBofcanthebeedgederive.edfromthefollowing
result:binatorialcomTheorem3.7.19(B’).Themaximalnumberoflatticepointsonanedgeofa
d-dimensionalreflexivesimplexis2td−1+1,withequalityattainedonlyforSQd.
ThisresulthasbeenobservedbyHaaseandMelnikovin[HM04]ford≤4
h.researcthisinitializedandHereoneshouldstatethefollowingrelatedresult(see[Fuj03,2.2]):
Proposition3.7.20.LetQbeanormalizedweightsystemwithassociatedsim-
plexPQ=conv(v0,...,vd).Leti,j∈{0,...,d},i=j.Thenwehaveforthe
torus-invariantintegralcurveConP(Q)associatedtothewallpos(vk:i=
k=j):
Q||(−KP(Q)).C=lcm(qi,qj).
pInartitionp(karticular,,...,k)when,weQget:isareflexiveweightsystemwithcorrespondingunit
d0(−KP(Q)).C=gcd(ki,kj).

3.7.Latticepointsinreflexivepolytopes87
ProofoftheoremB’.ThefactthatSQdsatisfiestheboundistrivialfromthe
3.6.18.anddefinitionLetP=conv(v0,...,vd)⊆MRbeareflexivesimplexwithassociatedweight
systemQP,andletQ:=(QP)red=(q0,...,qd)correspondtotheunitpartition
(k0,...,kd).Let[vd,vd−1]containthemaximalnumberoflatticepointsonan
edgeofP.Letm∈Mbetheprimitivelatticepointsuchthatvd−vd−1=cm
forc∈N>0.Obviously|[vd,vd−1]∩M|=c+1.
Assumec≥2td2−1.Sinceid=0qivi=0,wehavecηFd,m=ηFd,vd+1=
q|Qd|−1+1=kd.FurthermorecηFd−1,m=−kd−1.Hence
c|gcd(kd,kd−1).(3.9)
AnalternativeargumentwouldhaveusedProposition3.7.20.
Aswehavealreadyseenintheproofof3.6.29(takej=1)applyingLemma
yields3.6.30111kd−1+kd≥td−1.(3.10)
Fromnowonweassumewithoutrestrictionthatkd≥kd−1.Hencetheprevious
kd.Inparticularwegetdi−=021/ki=1−1/td−1.
twoequationsyield2td−1≤c≤kd−1≤2td−1,sowehavec=2td−1=kd−1=
NowLemma3.6.30implies{k0,...,kd−2}={y0,...,yd−2}.HenceQ=Qd
correspondstotheenlargedSylvesterpartition.
By3.6.18wecanchooseV(Pred)={e1,...,ed,e:=−q0e1−∙∙∙−qd−1ed}
forsomeZ-basise1,...,edofM.ByTheorem3.6.7thereis(uptounimodular
equivalence)amatrixH={hi,j}inHermitenormalformofdeterminantλP
suchthatH(e1)=v0,...,H(ed)=vd−1,H(e)=vd.Recallthataquadratic
matrixHisinHermitenormalform,ifitisalowertriangularmatrixwith
naturalnumbersascoefficientssuchthathi,j<hj,jfori>j.
Sincevd−1isprimitive,wegethd,d=1,sovd−1=ed.Becauseof
2t1(H(e)−ed)=m∈Mand2tqi=y1fori=0,...,d−2
d−1d−1i
wegetthefollowingd−1equations:
h1,1∈N
y0y0(h2,1+h2,2)∈N
y0y1...
y0∙∙∙yd−3(hd−1,1+∙∙∙+hd−1,d−1)∈N.
y0yd−2
Usingthefactthatgcd(yi,yj)=1fori=j,wededucebyinductionthatyi−1
divideshi,ifori=1,...,d−1.Hencewehave
mQ=td−1=y0∙∙∙yd−2≤h1,1∙∙∙hd−1,d−1hd,d=detH=λP≤mQ.
Now3.6.15(3)impliesP=∼(Pred)∗=SQd.

88

Chapter3.Reflexivepolytopes

3.7.3Countinglatticepointsinresidueclasses
hoTherewtoisgetanaeasysharpbmethoounddonthattheisnumboriginallyerofduelatticetopBatointsyrevinsome[Bat82asp,ecialLemmacases:1]
Wesimplycountlatticepointsmoduloanaturalnumberk:
Definition3.7.21.Fork∈Nwehavethecanonicalhomomorphism
αk:M→M/kM=∼(Z/kZ)d.

ForaconvexsetC⊆MRwithC∩M=∅oneeasilyseesthattheminimal
k∈N≥1suchthattherestrictionofαktoC∩Misinjectiveisjustthemax-
imalnumberoflatticepointsonanintersectionofCwithanaffineline.This
invariantminusoneiscalledthediscretediameterofCin[Kan98].
Lemma3.7.22.Letd≥2,PacanonicalFanopolytopeandB⊆∂P∩Mwith
|[x,y]∩M|=2forallx,y∈B,x=y,x∼y.Letsdenotethenumberof
centrallysymmetricpairsinB.Then
|B|≤2d+1−2,s≥|B|+1−2d.

Proof.Weconsidertherestrictionofα2toB.AsPiscanonical,thefibreof0
isempty.Usingtheassumptionitisalsoeasytoseethatthefibreofanon-zero
elementin(Z/2Z)dhasatmosttwoelements,andinthecaseofequalityit
consistsofonepairofcentrallysymmetriclatticepointsinB.Fromthisthe
boundscanbederived.

Forasemi-terminalcanonicalFanopolytopeP(seeDefinition3.2.3),the
setB=V(P)satisfiestheassumptionsoflemma3.7.22,soweimmediatelyget
asharpboundonthenumberofverticesofP.
InparticularwegetaresultthatwasproveninthecaseofasmoothFano
polytopein[Bat99,Prop.2.1.11]:
Corollary3.7.23.LetPbeaterminalFanopolytope.Then
|∂P∩M|=|V(P)|≤2d+1−2.
Ifequalityholds,thenPiscentrallysymmetric.Thisholdsfortheterminal
reflexived-dimensionalstandardlatticezonotopeZd=conv(±[0,1]d).
Theresultsin[Kas03]showthatZdiseventheonlyterminalFanopoly-
topewiththemaximalnumberofverticesford≤3.Howeverthecomputer
classificationofKreuzerandSkarkeyieldstwonon-isomorphicfour-dimensional
terminalreflexivepolytopeswith25−2=30vertices.Theyhavedifferent
volumes,butofcoursethesamenumberoflatticepoints!
Thesecondcasewherecountingmodulokworksistheclassofcentrally
symmetricreflexivepolytopesaswilldescribedintheverylastsectionofthis
thesis.

4Chapter

TFanoerminal3-foldsGorensteintoric

ductiontroInInthischapterthree-dimensionalGorensteintoricFanovarietieswithtermi-
vnalarietiessingularitiesareareparticularlyclassified.interesting,Theysinceconsistdueofto100themildisomorphismnatureofclasses.theirsingu-These
sectionlaritiesthey3.4).Fareorstillinstanceclosetointhe[Nam97w]ell-knoNamikwna18wasmoprovothedtoricthatFananoyFano3-folds3-fold(see
withTheGorensteinclassificationterminalusesthesingularitiescomisbinatorialsmoothabledescriptionbyaofflatthesevdeformation.arietiesas
fiedinthree-dimensional[Kas03]evenallterminalreflexivthree-dimensionalepolytopterminales.VFeryanorecenptlyolytopes,Kasprzykhoweverclassi-his
proofcomputerreliedhaspartlybeenonusedcomputerandanycalculations,calculationhaswherebineentheexplicitlyreflexivewrittencaseheredown.no
Fortheclassificationwefirstobservethatanythree-dimensionalterminalreflex-
ivwithepvolytoperticesea,hasb,asc,a+facetsb−ceitherforaalatticetrianglebasiswitha,vb,certices.Thisa,b,cmeansorageometricallyparallelogram
lothatcallythelikez1zsingularities2−z3z4app=0.earingThenareweatusemostpropconifoldertiesofthesingularities,projectioni.e,theymaploandok
somespecialrelationsamongthevertices.
ManyideasforthisapproachareduetoBatyrev.FurthermoreM¨ullergavein
hispresentedDiplomarbausefuleit[Mlist¨ul01con]advisedtainingbvyertices,Prof.Batfacetsyrevandsomefiguresofpreliminarythesepresultsolytopandes
basedonthecomputerdatabase[KS04b].Howeverhedidnotyetdescribean
effectiveapproach,theproofswerenotrigorousandthelistcontainedseveral
errors,e.g.,polytopesno.8.12and8.21(inhisnotation),aswellasno.8.7and
isomorphic.actuallyerew8.14,Inthefirstsectionofthischapterwediscussvariousnotionsofprimitive
wascollectionsimportanandtfortherelationsasclassificationintroducedofFbanoyBat4-foldsyrevdueinto[Bat91Bat],yrevsince[Bat99this]toandol
alsoHereweshoenormouslywhowthisinfluenceddatathecaninmethosomedofcasesbclassificationeusedtoachievedcompletelyinthiscdeterminehapter.
thepolytopeanditsrelations.

89

90

Chapter4.TerminalGorensteintoricFano3-folds

smoInoththeFanosecondpolytopsectionewandeshodefinewinthatthethisfirstpreciselysubsectiongivtheesthenotionsetofofaquasi-three-
dimensionalterminalreflexivepolytopes.Moreoverweintroducesocalledquasi-
primitivecollectionsandrelations,theseareespeciallysuitablefordescribing
quasi-smoothFanopolytopes.Inthesecondsubsectionwedefinethenotionof
asymmetricvertex,whereitsantipodalpointisalsoavertex,andofanadditive
vertex,thatisthesumoftwoothervertices.Weexaminetheirpeculiarprop-
ertiesreflexivewhenpolytopproesjectinginProp.alongsuc3.4.1,haasvwertexellasbytheusingresultsthelistinofProp.tw3.2.2(7)o-dimensionaland
3.3.1.Prop.ItInreliestheonthirdthenotionsectionofweangiveAS-ptheoint,proofi.e.,ofathevertexmainthatisclassificationbothsymmetrictheorem.
andthattheadditivpe.olytopIfenohassucathmostAS-poineighttvexists,ertices,thensowbeyuseProp.Prop.3.1.6we3.2.4(2)cantouseshothew
classificationofthree-dimensionalpropernonsingulartoricvarietieswithPicard
numdescribberedfivineor[Oda88less,whic1.34].hareOntheminimalotherinhandtheifsensethereofexistsequivanariantAS-pblooint,w-upsthenas
wecanuseProposition4.2.17inthesecondsectiontocompletelyclassifythe
polytopIntheebylastitssectionwquasi-primitivegiveethelistrelations.of100quasi-smoothFanopolytopeswith
someoftheirinvariants.MoreoverweshowhowtodeterminethePicardnumber
ofthecorrespondingtoricvarieties.
Summaryofmostimportantnewresultsofthischapter:
∙vTherearietiesarewithuptoterminalisomorphismsingularities100(Prop.three-dimensional4.3.2,p.101)GorensteintoricFano
∙spTheirondingPicardreflexivnumepbersolytopdepeend(Prop.only4.4.2,onp.the113)combinatoricsofthecorre-
∙Thesmallsetcorrespofspondingecialreflexivrelationsepolytopamongesitscanvbeerticesuniquely(Prop.4.2.17,determinedp.by101;a
102)p.4.3.3,Prop.

4.1Primitivecollectionsandrelations
impHereortanthetnotionsofalgebraic-geometricprimitivepropcollectionsertiesinandparticularrelationsareconcerningdescribtheed.assoForciatedtheir
and1-cycles[Cas03cin].theInMorithecaseconeofwaereferprimitivtoe[Rei83collection],[Bat91],consisting[BC94of],only[Bat99two],elemen[Sat00ts],
oneshouldcomparethedefinitionsandresultsherewithProposition3.3.1and
there.maderemarkstheForanycompletefanthefollowingdefinitionmaybeuseful.
Definition4.1.1.Letbeacompletefan.AsubsetP⊆(1)willbecalled
aprimitivecollection,ifitsatisfiestheconditions
1.Foranyσ∈wehavepos(P)⊆σ,
2.Foranyτ∈Pthereisaσ∈suchthatpos(P\τ)⊆σ.

4.1.Primitivecollectionsandrelations

91

Thisdefinitioncanalsobefoundonp.11ofthemanuscript[Cox03].When
thefanconsistsonlyofsimplicialcones,asetofraysiscontainedinaconeof
thefanifandonlyifitgeneratessomeconeofthefan.Hencethisdefinition
coincideswiththedefinitionofaprimitivecollectionasgivenin[Bat91]in
thecaseofanonsingularpropervariety.Bysubstituting⊆with=inany
oftheabovetwoconditions,wegetdifferentgeneralizationsofthenotionof
aprimitivecollection.Howeveronlythegivendefinitionyieldsthefollowing
ation:observtialessenLemma4.1.2.LetCbeasubsetofraysofacompletefan.ThenCis
containedinaconeofifandonlyifCdoesnotcontainaprimitivecollection.
Thisholds,sinceprimitivecollectionsarepreciselytheminimalelements
amongthesubsetsofrayswherenotallelementsarecontainedinacommon
cone.Fromthisresultweseethatprimitivecollectionssufficetodeterminethe
setofraysgeneratingmaximalcones.Sinceanyconeinacompletefanisthe
intersectionofmaximalcones,wederive:
Proposition4.1.3.Twocompletefansarecombinatoriallyisomorphic,i.e.,
thereisabijectionbetweenthesetofraysmappinggeneratorsofconesonto
generatorsofcones,ifandonlyifthereisabijectionbetweenthesetofrays
mappingprimitivecollectionsontoprimitivecollections.
FromnowonletP⊆MRbeareflexivepolytopeand:=ΣP.
Definition4.1.4.AsubsetP⊆∂P∩Mwillbecalledaprimitivecollection,
conditionsthesatisfiesitif1.ForanyF∈F(P)wehaveP⊆F,
2.Foranym∈PthereisaF∈F(P)suchthatP\{m}⊆F.
WhenP⊆V(P),PiscalledaV(P)-primitivecollection.
HenceV(P)-primitivecollectionsjustcorrespondtoprimitivecollectionsof
=ΣP.By4.1.3wegetthatthesetofV(P)-primitivecollectionsdetermines
thecombinatorialtypeofP.Forinstancethed-dimensionalcrosspolytopecan
beuniquelycombinatoriallydescribedas2dverticespartitionedintodpairsof
primitivecollections.Inthecaseofaterminalreflexivepolytopethenotionsof
V(P)-primitiveandprimitivecollectionsofcoursecoincide.
Asbeforewehave:
Lemma4.1.5.LetCbeasubsetof∂P∩M.ThenCiscontainedinafaceof
PifandonlyifCdoesnotcontainaprimitivecollection.
TodeterminePasalatticepolytopewegiveageneralizationofthenotion
ofaprimitiverelationasdefinedin[Bat91].
Definition4.1.6.LetP:={w1,...,wl}beachosenprimitivecollectionofP.
ThereisauniquefaceG(P)ofPsuchthat
lσ(P):=wl∈relint(pos(G(P))).
=1i

92

Chapter4.TerminalGorensteintoricFano3-folds

ThenbyHelly’stheorem1.5.1wefindanequation
tσ(P)=λjvj,
=1jwhereλj∈Q>0andv1,...,vtarelatticepointsonG(P)and1≤t≤d;thisis
calledaprimitiverelationassociatedtoP.Wehave{v1,...,vt}=∅,ifft=0,
iffσ(P)=0.
Althoughasintrinsicasthenotionofaprimitivecollection,thesetofprim-
itiverelationsassociatedtoPcontainsingeneralmorethanoneelement.How-
everthisambiguitycanberesolvedbyexplicitlychoosingatriangulationofthe
e:olytoppLetbeafixedcrepantsubdivisionofcorrespondingtoaQ-factorial
weaktoricFanovarietywithterminalsingularitiesasinProp.2.3.12.Inthis
casebychoosingv1,...,vtasthegeneratorsofaconein,thereisaunique
primitiverelationcalledtheprimitiverelationofPwithrespectto.
LetFbeafacetofPcontainingv1,...,vt,setνF:=−ηF.Sincenotall
w1,...,wlarecontainedinF,wehave
lt0<λj=νF,σ(P)=νF,wi∈Z≤l−1.
=1i=1jHencewecandefinethedegreeofaprimitiverelationas
tdeg(P):=l−λj∈N≥1.(4.1)
=1jUndersomeassumptionsthereisaconvenientpropertywhendetermining
primitiverelations(generalizing[Bat91,Prop.3.1]):
Lemma4.1.7.Letil=1wi=jt=1λjvjbeaprimitiverelationsuchthat
λj≥1forallj=1,...,t.
Then{w1,...,wl}∩{v1,...,vt}=∅.
Proof.Assumew1=v1.LetCbethesmallestconeinthatcontains
w2,...,wl.DefineCasthesmallestconeincontainingv1,...,vt,ifλ1>1,
orcontainingv2,...,vtotherwise.Then
tlwi=(λ1−1)v1+λjvj∈relintC∩relintC.
=2j=2iHenceC=C.LetCbeaconeofcontainingv1,...,vt.Thenw2,...,wl∈
C⊆Candw1=v1∈C,acontradiction.
Wehavethefollowingresult:
Proposition4.1.8.Assumeforanyprimitivecollectionwecanchooseaprim-
itiverelationsuchthatλj≥1.
IfthelatticepointsofafixedfacetofareflexivepolytopeParegiven,thenP
isdeterminedbythesetofprimitivecollectionsandchosenprimitiverelations.

4.1.Primitivecollectionsandrelations93
Proof.LetF∈F(P)andx1,...,xdbeasetoflinearlyindependentvertices
ofF.Letw∈∂P∩Mwitha:=νF,w≤0suchthatanyy∈∂P∩Mwith
νF,y>aisalreadydetermined.
Since{w,x1,...,xd}isnotcontainedinafaceofP,by4.1.5wecanassume
w+i=1xi=j=1λjvjtheassociatedprimitiverelation.IfνF,vi>afor
thereissaprimitivtecollectionP={w,x1,...,xs}(s≤d).Nowletσ(P):=
allj=1,...,twearefinished.t
SowematyassumeνF,v1≤a.tHencea+s=νF,σ(P)t=j=1λjνF,vj
≤λ1a+j=2λjνF,vj≤λ1a+j=2λj.Sinces+1−j=1λj≥1by(4.1),
yieldsthist0<λ1≤s−λj≤(λ1−1)a.
=2jSincea≤0,thisimpliesλ1<1,acontradiction.
IfAninanalogousDefinitionresult4.1.6cancanbebeformchosenulatedtobforeV(P)nonsingular,-primitivethenthecollections.coefficients
λjinaprimitiverelationwithrespecttoarenon-zeronaturalnumbers,in
particulartheysatisfytheconditionofthepreviousproposition.Hencethe
intrinsicandfinitelymanyconditionsthatforanyprimitivecollectionPthere
existlatticepointsinthefaceofPthatcontainsσ(P)intherelativeinteriorsuch
thatσ(P)isanon-negativeintegercombinationareimportantobstructionsfor
theexistenceofacrepantresolution.Howeverobviouslytheyarenotsufficient
ascanbeenseeninExample3.2.6.
Asacorollarywegeneralizeawell-knownresultfornonsingulartoricFano
]):[Bat99(seearietiesvacrepCorollaryantr4.1.9.esolutionAisreflexiveuniquelypolytopdetermineecorrdespbytheondinglattictoeaptoricointsofvarietyonearbitradmittingary
facetandthesetofprimitivecollectionsandprimitiverelations(chosenwith
efficients).ocgerinteWecanalsodeterminethegroupoflinearrelations:
theDefinitiongroupoflinear4.1.10.LetrelationsC⊆withMRinbetegeracolatticeefficienptsolytope.amongWeletelemenLRts(Cof)C∩denoteM.
Proposition4.1.11.Assumeforanyprimitivecollectionwecanchoosea
primitiverelationsuchthatλj∈N.
ThenthegroupLR(∂P)isgeneratedby{LR(F):F∈F(P)}andthe
elations.rprimitivechosenProof.Letsr
αiwi−βjuj=0
=1j=1ibearelationgsuchthat{αi},{βj}arenon-zeronaturalnumbers,W:=
{w1,...,ws}⊆∂P∩MandU:={us1,...,ur}r⊆∂P∩MwithV∩W=∅,
theandtheproperty’absolutethatthisnorm’inabs(tegerg):=relationig=1isαinot+conj=1βtainedj∈inN>the0issubgroupminimalHwithof
LR(∂P)generatedby{LR(F):F∈F(P)}andthesetofprimitiverelations.

94

Chapter4.TerminalGorensteintoricFano3-folds

IfWandUareeachcontainedinafaceofP,thentheirrelativeinteriors
intersect,henceWandUarecontainedinacommonfacetofP,howeverthisis
acontradictiontog∈H.SowemayassumeWisnotcontainedinafaceofP,
soby4.1.5wecanassumethatthereisaprimitivecollectionP={w1,...,wl}
(l≤s)andacorrespondingprimitiverelation
tlwi−λjvj=0
=1j=1iwithλj∈N>0.Thedifferenceoftheprevioustwoequationsyieldsarelationg
lstr
(αi−1)wi+αiwi+λjvj−βjuj=0.
i=1i=l+1j=1j=1
Necessarilyg∈H.Regroupingtheelementsyields
abs(g)≤(αi−1)+αi+λj+βj=abs(g)−deg(P)<abs(g),
lstr
i=1i=l+1j=1j=1
by(4.1),acontradiction.

4.2Combinatoricsofquasi-smoothFanopoly-
estop4.2.1Definitionandbasicproperties
Definition4.2.1.Letd=3andFasubsetofMR.WecallFasmooth
triangle,ifF=conv(v1,v2,v3)suchthatv1,v2,v3isaZ-basisofM.
Definition4.2.2.Letd=3andFasubsetofMR.WecallFasmooth
latticeparallelogram,ifthereexistlatticeelementsv1,v2,v3,v4∈Mhaving
Fasitsconvexhullsuchthatv1,v2,v3formaZ-basisofthelatticeMand
v1+v3=v2+v4.Inthiscasev1,v2,v3,v4aretheverticesofthelattice
polytopeFandanysubsetofitsverticeshavingthreeelementsformaZ-basis
ofthelatticeM.[v1,v3]and[v2,v4]arecalledthediagonalsoftheparallelogram.

Definition4.2.3.Letd=3andPapolytopeinMRwith0∈intP.WecallP
aquasi-smoothFanopolytope,ifeachfacetFofPiseitherasmoothtriangle
orasmoothlatticeparallelogram.
PhasonlytrianglesasfacetsiffitisasmoothFanopolytope,orequivalently
X(M,ΣP)isnonsingular.OtherwisePcontainsatleastoneparallelogramas
afacet,wecallPsingularinthiscase.
Hereisthemotivationforthisdefinition:
Proposition4.2.4.Thesetofquasi-smoothFanopolytopesispreciselytheset
ofthree-dimensionalterminalreflexivepolytopes.
Isomorphismclassesofthree-dimensionaltoricFanovarietieswithatmost
conifoldsingularitiescorrespondtoisomorphismclassesofGorensteintoric
Fanovarietieswithterminalsingularities.

4.2.Combinatoricsofquasi-smoothFanopolytopes95
Proof.Obviouslyanyquasi-smoothFanopolytopeisaterminalreflexivepoly-
tope.terminal,OnantheyfacetotherofhandPisletanPbempteaytwterminalo-dimensionalreflexiveplatticeolytoppe.olytopSincee.ItPisis
elementarytoseethatanysuchtwo-dimensionalpolytopeislatticeequivalent
toconv(0,e1,e2)orconv(0,e1,e2,e1+e2)foratwo-dimensionallatticebasis
e1,e2.SincePisreflexive,3.1.8(2)showsthatPisquasi-smooth.
Lete1,e2,e3beaZ-basisofM.Ifσ=pos(e1,e2,e3),thenwegetσ∨=
pos(e1∗,e2∗,e3∗).Ifσ=pos(e1,e2,e3,e1+e3−e2),thenσ∨=pos(e1∗,e3∗,e1∗+
e2∗,e2∗+e3∗).Inparticulartheonlysingularitiesofatoricvarietycorresponding
tothefanspannedbyaquasi-smoothFanopolytopeareisolatedsingularities
ofthetypez1z3−z2z4=0.
yields:3.7.22LemmaApplyingFPropanopositionolytopeP4.2.5.isnotThegreaternumberthanof14.verticesThen=numbn(erP)ofofanysymmetricquasi-smopairsothof
verticesisatleastn−7.
LetP⊆MRbeareflexivepolytopeand:=ΣP.
SincebyProp.3.1.6anythree-dimensionalreflexivepolytopehasacrepant
resolution,by4.1.8itisdeterminedbytheverticesofafacet,primitivecollec-
tionsandprimitiverelationswithintegercoefficients.Forourpurposesitwill
makeevensensetodefineanothertypeofprimitivecollection:
Definition4.2.6.AsubsetP⊆∂P∩Mwillbecalledaquasi-primitivecol-
lection,ifitsatisfiestheconditions
1.ForanyfaceGofPwehaveconv(P)=G,
2.Foranym∈PthereisafaceGofPsuchthatconv(P\{m})=G.
Quasi-primitivecollectionsoforder≥2arepreciselytheminimalelements
amongHencethewesubsetsonlyofgetV(oneP)thatdirectionareofnottheLemmavertex4.1.5:setAofnysomesubsetfaceofofV(PP.)that
isnotafaceofPcontainsaquasi-primitivecollection.Sinceaquasi-primitive
collectionthatisnotcontainedinafacetisalsoaprimitivecollection,there
arefewercollectionsofthiskind.Ontheotherhand,whenaquasi-primitive
asincollection(4.1)isisconzero.tainedSoinmostaface,resultstheofthedegreeofpreviousacorrespsectionondingcannotbprimitiveegeneralized.relation
Wecan’artificially’makesenseofthenotionofaquasi-primitiverelationby
hold:to4.1.7LemmaurgingDefinition4.2.7.LetPbeareflexivepolytopeandP={w1,...,wl}aquasi-
primitivecollection.Ifwecanfindanequation
tlσ(P):=wi=λjvj,
=1j=1iwhereλj∈Q>0,andv1,...,vtarethegeneratorsoftheuniquefacewhose
positivehullcontainsσ(P)initsrelativeinterior,1≤t≤dand{w1,...,wl}∩
{v1,...,vt}=∅,thenitiscalledaquasi-primitiverelationassociatedtoP.

96Chapter4.TerminalGorensteintoricFano3-folds
Ingeneraltheseconditionsmightbetoostrong,sonoquasi-primitiverelation
canbeassociatedtoagivenquasi-primitiverelation.Howeverinthecaseofa
quasi-smoothFanopolytopewehaveexistenceanduniqueness:
Proposition4.2.8.LetPbeaquasi-smoothFanopolytopewith|V(P)|>4,
andP={w1,...,wl}aprimitive(respectivelyquasi-primitive)collection.
Thenexactlyoneprimitive(respectivelyquasi-primitive)relationcanbeas-
sociatedtoP.Therearethefollowingpossiblecases:
1.Forl=2:
(a)w1+w2=0
(b)w1+w2=v1forv1∈V(P),inthiscasew1∼v1∼w2
(c)w1+w2=v1+v2wherew1,w2,v1,v2aretheverticesofaparallelo-
gramfacet,thisistheonlycasewhereP={w1,w2}iscontainedin
acommonface,soPisnotaprimitivecollection.
2.Forl=3:
(a)w1+w2+w3=0
(b)w1+w2+w3=cv1forc∈{1,2}andv1∈V(P).
Proof.Letl=2.Thefourcasesfollowimmediatelyfrom3.3.1,terminalityof
Pandtheparallelogramrelation.Letl=3.Wecanassumethat{w1,w2,w3}
isnotcontainedinafacet.By(4.1)itisenoughtoassumetherewerevertices
v1,v2inafacetFofPsuchthatw1+w2+w3=v1+v2.Thennecessarily
ηF,w1+ηF,w1+ηF,w1=−2,sobyreflexivitywecanassumew1,w2∈
FandηF,w3=0.Sobydefinition{w1,w2,v1,v2}aretheverticesofa
parallelogramfacet,hencetheysatisfytheparallelogramrelation,fromthis
aneasycalculationyieldsacontradiction.l≥4wereonlypossibleforthe
smoothFanopolytopethatisthesimplexspanningthefancorrespondingto
space.ejectivprothree-dimensionalIn4.2.16thecase(1b)willbefurtherinvestigated.
Sinceanyprimitiverelationisnecessarilyintegral,wegetasacorollaryfrom
4.1.11:Corollary4.2.9.Thegroupoflinearrelationswithintegercoefficientsamong
theverticesofaquasi-smoothFanopolytopeisgeneratedbytheparallelogram
relationsoffacetsandthesetofuniqueprimitiverelationsassociatedtotheset
ofprimitivecollections.
Whenlookingatathesimplestsingularquasi-smoothFanopolytope,namely
thepyramidewithparallelogrambasisconv(e1,e2,−e1+e2+e3,e3)andapex
−e2−e3,weseethatthereisonlyonequasi-primitiverelation,i.e.,theparallel-
ogramrelation,howeverthegroupoflinearrelationshasrank2,sotheprevious
corollarydoesnotholdforquasi-primitiverelations.
Fromthisexampleweseethatitisnotatallobviousthatthesetofquasi-
primitiverelationsshouldsufficetodeterminetheisomorphismtypeofaquasi-
smoothFanopolytopeasalatticepolytope.Howeverbytheclassification4.3.2
thisresultholds,inparticularseeProposition4.2.17forapartialexplanation.

4.2.Combinatoricsofquasi-smoothFanopolytopes97

4.2.2Projectionsofquasi-smoothFanopolytopes
LetPbealwaysaquasi-smoothFanopolytopeandvi∈V(P)avertexofP.
Definition4.2.10.Letπi:=πvi,Pi:=πi(P),Mi:=Mvi,ιi:=ιviasin3.2.2.
Let∂M(vi):=∂(vi)∩M=(st(vi)∩V(P))\{vi}.
Thenumberdeg(vi):=|∂M(vi)|ofvertices=viinthestarsetofviiscalled
thedegreeofvi.
Twoelementsvj,vk∈∂M(vi)arecalledcontiguouswithregardtoπi,if
πi(vj),πi(vk)arecontainedinafacetofPisuchthat[πi(vj),πi(vk)]∩Mi=
{πi(vj),πi(vk)}.
Itisimportanttoremarkthatitispossibletodeterminethestarsetand
therebythedegreeofavertexbyknowingonlythequasi-primitivecollections
andquasi-primitiverelationsofPasfollowsimmediatelyfromthedefinition
4.2.8(1).andSinceby3.1.8canonicalFanopolygonsarereflexive,wegetbyapplying
3.2.4(2)and3.2.2thefollowingresult:
Proposition4.2.11.Athree-dimensionalreflexivepolytopeisaquasi-smooth
Fanopolytopeifandonlyiftheprojectionalonganyvertexyieldsacanonical
e.olytoppanoFWehavethefollowingproperties:
1.Piisatwo-dimensionalreflexivepolytope.
2.If0=w∈Mi∩Pi,thenπi−1(w)∩V(P)={vj,vk},withvj=vk∈∂M(vi)
orvj+vi=vk∈∂M(vi)orvk+vi=vj∈∂M(vi).
3.Theelementsof∂M(vi)canbe(uptoreversionandcyclicpermutation
uniquely)orderedassl+1=s1,s2,...,slwithl=deg(vi)suchthatsj,sj+1
arecontiguousforj=1,...,l;thenconv(vi,sj,sj+1)iscontainedina
facetofPforj=1,...,l;if[vi,sj]isnotanedge,thennecessarily
conv(vi,sj−1,sj,sj+1)isasmoothparallelogramfacet.
Thismotivatesthefollowingdefinition:
Definition4.2.12.Anonzerolatticepointw∈Piiscalleddoublepointwith
regardtoπi,ifthereexisttwodifferentverticesvj,vk∈V(P)suchthatπi(vj)=
w=πi(vk).Otherwisewiscalledsimplepointwithregardtoπi.
By4.2.11itispossibletodeterminethedoublepointsofπibyknowingonly
thequasi-primitivecollectionsandquasi-primitiverelationsofP.
Asanotherapplicationof3.2.2thereisthefollowingobservation(dueto
Batyrev,see[M¨ul01]).
Proposition4.2.13.LetΓbeafacetofPi.ThenG:=πi−1(Γ)∩Pisaface
ofP,wehave|Γ∩Mi|≤3.
1.If|Γ∩Mi|=2,thenGisanedge,atriangleoraparallelogram,respective
Γhavingno,oneortwodoublepoints.
2.Let|Γ∩Mi|=3andΓ∩Mi={x,y,z}withy=(x+z)/2.ThenGisa
facet,[vi,ιi(y)]isanedge,andx,zaresimplepoints.Gisaparallelogram
iffyisadoublepoint.

98Chapter4.TerminalGorensteintoricFano3-folds
InparticularthereareuptoisomorphismofthelatticeMiexactly11possible
typesofπi-imagesofPasgiveninthefollowinglistorderedbythedegreeofvi.
Linesbetween0andalatticepointwdenotethat[vi,ιi(w)]necessarilyhasto
beanedgeofP.

4c4b34a

6b6a5b5a

8a7a6cProof.Sinceby3.2.2andterminalityofPthesetofverticesofGmapssurjective
onthesetoflatticepointsinΓ,theedgeΓhasatmostfourlatticepoints.By
4.2.11(2)thecase|Γ∩Mi|=2istrivial.
Let|Γ∩Mi|≥3.Thenby3.2.2(6)Ghastobeatriangleoraparallelogram
facetthatisparalleltovi.By3.2.2(2)thesetofverticesofGinthestarsetofvi
isinbijectionwiththesetoflatticepointsinΓ.However|∂M(vi)∩V(G)|=3.
IfGisaparallelogram,thenbydefinitionobviouslythemiddlepointyhasto
beadoublepoint.
withF[orvi,vthej]listnotusean3.4.1edge,andthenπi(4.2.11(3),vj)isandadditiveobservbyethethat,ifvjparallelogram∈∂M(vi)∩relation.V(P)
TherearethreespecialclassesofverticesofP:
Definition4.2.14.viiscalledadditive,ifviisthesumoftwootherverticesofP,i.e.there
existsaquasi-primitiverelationvj+vk−vi=0forverticesvj,vk∈V(P).
viiscalledsymmetric,if−vi∈V(P)isalsoavertex,i.e.thereexistsa
quasi-primitiverelationvi+vj=0foravertexvj∈V(P);
viiscalledAS-point,ifviisadditiveandsymmetric.
comInthebinatoricssymmetricofPandcasetherequasi-primitivismoreetosaycollectionsabouttheamongdoubleelemenptsoinofts∂ofMπ(iv,i).the
Lemma4.2.15.Letvi∈V(P)beasymmetricvertexofP.
1.Vof(PP)=wher(ste(πvii()v∪k)st(is−avi))∩simpleV(Pp);oint;∂M∂(vMi)(v∩i∂)\M∂(M−(vi−)vic)containsontainsthetheverticverticesvesk
vvkk−ofvPi∈∂wherMe(−πiv(i)v\k)∂Mis(avi).doubleApvertexoint,intheV(Ppr)eimagewhoseispr{ojevk,ctionvk−isvia},doublewhere
additive.isointp

4.2.Combinatoricsofquasi-smoothFanopolytopes99

2.AnyfacetGofPeithercontainsvior−viasavertexoristheinter-
sectionofPandtheπi-preimageofafacetΓofPi.Ifinthelastcase
|Γ∩Mi|=3,thenGisaparallelogram(themiddlepointofΓ∩Miisa
doublepoint,theothertwolatticepointsaresimplepoints).Wehavefor
vk,vl∈∂M(vi):
[vk,vl]isanedgeofPifandonlyifvk,vlarecontiguous.
e:eforTher{vk,vl}isaquasi-primitivecollectionifandonlyifvk,vlarenotcontigu-
ous.Ifinthiscasevk∼vl,then[vk,vl]isadiagonalofaparallelogram
facet(notnecessarilycontainingvi),sothereisanothervertexvt∈∂M(vi)
contagioustobothvkandvl.
3.Letvk,vl∈∂M(vi)betwoverticeswithπi(vk)=−πi(vl).Then{vk,vl}
isaquasi-primitivecollectionwithvk+vl∈{vi,0,−vi}.
Ifπi(vk)andπi(vl)aredoublepoints,thenvk+vl=vi.
Ifvk+vl=−vi,thenπi(vk)andπi(vl)aresimplepoints.
Ifvk+vl=0,thenvior−viisanAS-point.
Ifπi(vk)isadoublepoint,thenvkorvk−viisanAS-point.
Thesestatementscanbeeasilyverifiedusing3.2.2(7),4.2.11,4.2.8.
Intheadditivecasetherearestrongrestrictionsonthestructureofst(vi):
Lemma4.2.16.Letvi∈V(P)beanadditivevertexofPwithvj+vk=vifor
verticesvj,vk∈V(P).
Thereareexactlytwoverticesvl,vr∈∂M(vi)\{vj,vk}suchthat[vi,vl]and
[vi,vr]areedgesofP.
Preciselythefollowing10casescanoccur(uptoexchangingofvjandvk,
respectivelyvlandvr),wherethelabelsdenotethedegreeofvi:
1.[vi,vj]and[vi,vk]areedges:

vvvjvrjvrjvr
vivivi
vlvlvl
vkvkvk
465

vvvjvrjvrjvr
vivivi
vlvlvl
vkvkvk
676

100

Chapter4.TerminalGorensteintoricFano3-folds

vjvrvivlvk82.Either[vi,vj]or[vi,vk]isandiagonal(herewithoutrestriction[vi,vk]):
vjvjvj
vivrvivrvivr

vvkvvkvvk
lll654

Proof.SincePisathree-dimensionalpolytope,thereareatmosttwofacets
containing[x,y]forverticesx,yofP,withequalityiff[x,y]isanedge.
SoletFbeafacetofPcontainingvi.By4.2.8Fcontains[vi,vj]or[vi,vk].
Sothereareatmostfourfacetscontainingvi,especiallyanyvertexinthestar
setofvihastobeavertexofoneofthesefacets.
Assume[vi,vj]and[vi,vk]werediagonals,i.e.,thereareexactlytwofacets
containingvi.Thisimpliesthatvj,vr,vk,vlisanorderingof∂M(vi)asin
4.2.11(3),whereconv(vi,vj,vl,vr)andconv(vi,vk,vl,vr)areparallelograms.
Necessarilyvi+vj=vl+vrandvi+vk=vl+vr,anobviouscontradiction.
3.3.1(iv).useelyAlternativ

Combiningtheprevioustwopropositionswegetaresultthatwillbethekey
factorinclassifyingquasi-smoothFanopolytopes:
Proposition4.2.17.Quasi-smoothFanopolytopeshavingAS-pointsareuni-
quelydeterminedbytheirquasi-primitiverelations.
Morepreciselythefollowingdatauniquelydeterminesthequasi-smoothFano
:Peolytopp1.TheexistenceofanAS-pointvi∈V(P)(withquasi-primitiverelation
vj+vk=vi)
2.Theassociatedquasi-primitiverelationofthequasi-primitivecollection
{vl,vr}fortheuniqueverticesvl,vr∈V(P)\{vj,vk}suchthat[vi,vl]and
[vi,vr]areedges
3.Whatcasein4.2.16occurs,i.e.whichofthelines[vj,vr],[vj,vi],[vj,vl],
[vk,vr],[vk,vi],[vk,vl]arenodiagonals
4.WhichoftheboundarylatticepointsinPiaredoublepoints

4.3.Classificationofquasi-smoothFanopolytopes101

Theverticesoccuringinthequasi-primitiverelationassociatedto{vl,vr}
arecontainedin
{vl,vj,vj−vi,vi,0,−vi,vr,vk,vk−vi}.

Proof.Letviasin1.By4.2.16vl,vrasin2.existandarenotcontagious,so
{vconl,vr}tainedisainafacequasi-primitivofP,soeitiscollectionpossiblebytoc4.2.15(2).hoosevrBy,vj,v4.2.16iasaconZv(vi-basis,vj,ofvr)theis
lattice.Obviously−viandvkaredeterminedbyvr,vj,vi.Nowtheadditional
statementabouttheverticesoccuringinthequasi-primitiverelationof{vl,vr}
followsfromanalyzingthepossibletypesofPiin4.2.13andusing4.2.8and
4.2.11.Sovlcanbedeterminedbyvi,vj,vrandtherelationin2.By3.and
4.2.16allverticesinthestarsetofviaredeterminedbyvi,vj,vk,vr,vl,and
thereforebyvr,vj,vi.Nowtakeavertexv∈V(P),v∈{vi,−vi},thenby
4.2.11(2)thereisauniquevertexvh∈∂M(vi),andby4.itisknownwhether
v=vhorv=vh−vi.ThereforeallverticesofPareuniquelydeterminedby
1.-4.uptoanisomorphismofthelattice,andobviouslytheconditions1.to
4.aredeterminedbyquasi-primitiverelations.

4.3Classificationofquasi-smoothFanopolytopes
theoremmainThe4.3.1Thegoalofthissectionistoprovethefollowingtheorem:
Theorem4.3.1.Thereexistexactly100isomorphismtypesofthree-dimensional
GorensteintoricFanovarietieswithterminalsingularities.Oftheseare18non-
singular.82andsingularThisisacorollaryofthefollowingconvex-geometricformulation:
Theorem4.3.2.Thereexistexactly100isomorphismtypesofquasi-smooth
Fanopolytopes.Oftheseare18smoothand82singular.Quasi-smoothFano
polytopesareuniquelydeterminedbythequasi-primitiverelationsassociatedto
theirquasi-primitivecollections.
numberofvertices4567891011121314
numberofpolytopes1511182318136311
ofthesearesmooth14742000000
ofthesearesingular014142118136311

Thestrategyoftheclassificationprocessisbasedupontheobservationsin
theprevioussection.Thereareessentiallytwodifferentparts:
Ifaquasi-smoothFanopolytopePhasanAS-point,thenapplyingthekey
resultProposition4.2.17givesusanexplicitalgorithmforuniquelydetermining
P.Thisisdoneinsubsection4.3.3.

102Chapter4.TerminalGorensteintoricFano3-folds

caseOnwhenthethereotherarehandnoifnosymmetricAS-poinvtertices,ofPhereexists,wecanthenwapplyefirstandealexistingwithclas-the
v,sificationthenweofuseOdatheinfact[Oda88that].Iftherethereareonlyexists11notypAS-pesofoinptbutolytopaesthatsymmetriccanvoertexccur
whenprojectingalongvtoproveverystrongrestrictionsonthepolytopethat
willsufficetodetermineP.
4.3.2ClassificationwhennoAS-pointsexist
Inthissubsectionwearegoingtoprovethefollowingproposition:
Proposition4.3.3.Thereexistexactly17quasi-smoothFanopolytopeshaving
3notheAS-pnotationoints.ofOf[Bat99,thesear2.5])e8andsmo9othsingular(namelyP(namely,B1,5.1,B2,6.2,B3,B6.4,4,C37.2,,D17.3,,D27.11,in
7.13,8.2,8.21).Theyareuniquelydeterminedbythequasi-primitiverelations
ofassotheciate9dtosingulartheirpolytopesquasi-primitivearegivencolrleelativections.toaInZthe-basisfolv,lowingv,vlistofthetheverticlattice:es

n0verticesofP
5.1v,v,v,−v−v,−v−v
6.2v,v,v,−v−v,−v−v,v+v+v
6.4v,v,v,−v,−v,−v+v+v
7.2v,v,v,−v,v−v,v−v,−v−v
7.3v,v,v,−v,v−v,v−v,−v−v+v
7.11v,v,v,−v,−v,−v,−v−v+v
7.13v,v,v,−v−v,−v−v,−v−v,−v−v−v
8.2v,v,v,−v,v−v,v−v,−v−v+v,−v−v,
8.21v,v,v,v−v+v,−v,−v,−v,−v+v−v
Thefirstcaseishandledusingtheexistingclassificationofnonsingular
propertoricvarietieswithaverysmallPicardnumber:
Lemma4.3.4.Thereexistexactlythreequasi-smoothFanopolytopesthatare
singularandhavenosymmetricvertices.Theseare5.1,6.2,7.13.
Proof.By4.2.5thenumberofverticesofPissevenorless.Aslistedin[Oda88,
Thm.1.34]thereareexactlythreethree-dimensionalcompactnonsingulartoric
varietieswithPicardnumberfourorlessbeingminimalinthesenseofequivari-
antblowing-upswheretheassociatedfanhasnosymmetricpairofgeneratorsof
one-dimensionalcones.Inthenotationgiventheretheseare34,(3243),314353.
Thelastfancorrespondsalreadytoaquasi-smoothFanopolytopethatissin-
gular,namely7.13.Nowonehasjusttocheckthatbyequivariantlyblowing-up
thefirsttwovaritiesatmostthree,respectivelytwotimesonecanonlyget5.1,
apyramidewithaparallelogrambasis,or6.2,astackedsimplexonapyra-
mide,asassociatedquasi-smoothFanopolytopesthataresingularandhaveno
symmetricvertices.Thisisaneasybuttediouscalculationthatwillbeomitted.

Thenextlemmawillgiverestrictionsonquasi-smoothFanopolytopesto
havenoAS-points:

4.3.Classificationofquasi-smoothFanopolytopes103
Lemma4.3.5.LetPbeaquasi-smoothFanopolytopehavingnoAS-point.Let
vi∈V(P)beasymmetricvertex.Ifthereexistw1,w2nonzerolatticepointsin
Piwithw1+w2=0,thenw1andw2aresimplepoints.Letv1,v2∈V(P)be
theuniqueverticeswithπi(v1)=w1andπi(v2)=w2.Thenv1+v2=0.
Proof.BecausePhasnoAS-pointsitfollowsfromthestatementsin4.2.15(3)
thatv1+v2=0andw1,w2aresimplepoints.

Nowtheproofofthepropositioncouldalsobedonebytheclassification
ofOda,becauseintheproofitwillbeshownthatn(P)≤8isanecessary
conditionforaquasi-smoothFanopolytopetohavenoAS-points.Insteadthe
polytopeswillbeclassifiedusingthelanguageofquasi-primitiverelationsas
thisseemstobemoreinstructive.
ProofofProposition4.3.3.By[Bat99,2.5]itiseasytocheckthatthereexist
exactly8smoothFanopolytopeshavingnoAS-points.SoletPbesingular.
By4.3.4wecanassumethatthereexistverticesvi,vj∈V(P)withvi+vj=0.
By4.2.15(2)afacetofPihavingthreelatticepointscontainsthemiddlepoint
astheonlydoublepoint.Alsoby4.3.5symmetricnonzerolatticepointsofPi
aresimplepointsandtheiruniquepreimagesinV(P)aresymmetric.Looking
upall11possibleprojectiontypesin4.2.13oneseesthatthereareonlythe
left:caseswingfollo3:InorderforPnotbeingsimplicialtherehavetoexistatleasttwodouble
pointsofPi.
IftheremainingnonzerolatticepointofPiisnodoublepoint,thenonecan
chooseaZ-basise1,e2,e3=viofMsuchthatPconsistsofaFanopyramide
withbasise1,e2,e1−e3,e2−e3andapexv∈V(P)withπi(v)=(−1,−1)and
twostackedsimpliceswithverticesv,e3,e1,e2,respectivev,−e3,e1−e3,e2−e3.
By4.2.8(2)v+e1+e2∈{0,e3,2e3}andv+(e1−e3)+(e2−e3)∈{0,−e3,−2e3}.
Bysymmetrythisgivestwonon-isomorphicquasi-smoothFanopolytopeswith
sevenverticesdependingonwhetheroneofthesetwoquasi-primitiverelations
equals0;ifonedoes,Pisisomorphicto7.2,otherwise7.3.
IftheremainingnonzerolatticepointofPiisadoublepoint,thenone
showssimilarilythatPisevenuniquelydeterminedasaquasi-smoothFano
polytopewitheightverticesconsistingofaprismawithtwostackedsimplices;
8.2.toisomorphicisP4a:Becauseby4.3.5Pthenconsistsofsixsymmetricvertices,Pwouldbe
anoctahedron,whichissimplicial.Acontradiction.
4b:By4.3.5thereexistvj,vk∈V(P)withvj+vk=0,πi(vj),πi(vk)simple
pointsandvl,vr∈∂M(vi)withπi(vl+vr)=πi(vk).By4.2.15{vl,vr}isaquasi-
primitivecollection.Assumevl+vr∈V(P),thenvl+vr=vk,becauseπi(vk)
isasimplepoint.ThismeansvkisanAS-point,acontradiction.Therefore
[vl,vr]isadiagonal,by4.2.15vl+vr=vi+vkwithoutrestriction(substitute
−viforviotherwise).Assumeπi(vl)wereadoublepoint.Thenvl−vi∈V(P)
and(vl−vi)+vr=vk,contradiction.Thesameistrueforπi(vr).Sothere
arenodoublepointsinPiandallsixverticesinPareuniquelydefined.Pis
6.4.toisomorphic5a:By4.3.5takeanorderingvj,vr,vt,vk,vlof∂M(vi)suchthatvj+vk=0
andvl+vr=0andallpointsexceptpossiblyπi(vt)aresimplepointsofPi.

104Chapter4.TerminalGorensteintoricFano3-folds

Exactlyasinthepreviouscaseonecanassumethatvl+vt=vi+vkandπi(vt)
isasimplepoint.Thisyields7.11.
cause6a:PBycon4.3.5tainsPaissmoaothcompletelylatticepolytopsymmetrice,bypolytopsymmetryewithitfolloeighwstvertices.immediatelyBe-
thatPistheuniquequasi-smoothFanocube8.21.
Finallyitiseasytocheckthatallthesequasi-smoothFanopolytopesare
uniquelydeterminedbytheirquasi-primitiverelations.

4.3.3ClassificationwhenAS-pointsexist
Inthissubsectionthefollowingpropositionwillbeprovedwhichwillyieldto-
getherwithProposition4.3.3theproofofTheorem4.3.2:
Proposition4.3.6.Thereexistexactly83quasi-smoothFanopolytopeshav-
inganAS-point.Oftheseare10smoothand73singular.Theyareuniquely
determinedbythequasi-primitiverelationsassociatedtotheirquasi-primitive
ctions.leolcThesecondpartandthemainideaoftheproofiscontainedinthekeyresult
4.2.17.ositionPropNowtheclassificationofsingularquasi-smoothFanopolytopesthathavean
AS-pointissplitupintofourLemmas4.3.9to4.3.14dependingontheminimal
degreeofanAS-point.Fromtheproofsoftheselemmasthecoordinatesofthe
verticesofthepolytopescanbeimmediatelyreadoffbychoosingvr,vj,vi(as
in4.2.17)asaZ-basisofthelattice.
LetPbealwaysaquasi-smoothFanopolytope.
Tosimplifytheproofsweneedtwodefinitions:
Definition4.3.7.Letvi∈V(P)andv∈V(P).
seteW∙mi(v):=|πi−1(πi(v))∩V(P)|.
Forv∈∂M(vi)weget:
mi(v)=1,iffπi(v)isasimplepoint,andmi(v)=2,iffπi(v)isadouble
point.Hence4.2.17(4)meansexactlytodeterminemi(v)forallv∈
∂M(vi).
∙Assumeviissymmetric.Thenthereisauniquevertexbi(v)∈∂M(−vi)
suchthatπi(bi(v))=πi(v).Forv∈∂M(vi)weget:
Ifmi(v)=1,thenbi(v)=v;otherwisebi(v)=v−viby4.2.15.
Thefollowingtechnicallemmathatwillbesubsequentlyusedshowshowto
deduceinsomecasesthecombinatoricsofst(−vi)fromst(vi).
Lemma4.3.8.Letvi∈V(P)beasymmetricvertexofP.Alsoletvl,vj,vr∈
∂M(vi)suchthatvl,vjandvj,vrarecontiguous.Weset
x:=(mi(vl),mi(vj),mi(vr))∈{0,1}3.
1.Let[vi,vj]beanedgeofPandvl+vr=vjaquasi-primitiverelation.
Thenx=(2,1,2).conv(−vi,bi(vl),bi(vj),bi(vr))isaparallelogramfacet
ofPiffx∈{(1,1,2),(2,1,1),(2,2,2)}.

4.3.Classificationofquasi-smoothFanopolytopes105
2.Let[vi,vj]beanedgeofPandvl+vr=vj−viaquasi-primitiverelation.
Ifmi(vj)=1,thenx=(1,1,1)andconv(−vi,vl,vj,vr)isaparallelogram
.PofetfacIfmi(vj)=2,thenx=(2,2,2).conv(−vi,bi(vl),vj−vi,bi(vr))isa
parallelogramfacetofPiffx∈{(2,2,1),(1,2,2)}.
3.Letπi(vl)+πi(vr)=πi(vj).Thenx=(2,1,2)iffconv(vi,vl,vj,vr)and
conv(−vi,bi(vl),bi(vj),bi(vr))areparallelogramfacetsofP.
4.Letconv(vi,vl,vj,vr)beaparallelogramfacetofP.Thenthereexistsno
quasi-primitiverelationvt+vl=vr−viforavertexvt∈V(P).
Proof.1.,2.,3.areeasyconsequencesof4.2.8,4.2.11and4.2.15,soaproofwill
omitted.eb4.Assumevt+vl=vr−viwereaquasi-primitiverelation.Ifmi(vr)=
1,thenconv(−vi,vt,vr,vl)wereaparallelogramfacetcontaining−vi,soby
4.2.11(3)vlwouldbecontagioustovr;acontradiction.Soletmi(vr)=2,then
by4.2.8vl∈st(vr−vi).By4.2.15(2)itfollowsthat[vl,vr−vi]isadiagonalof
aparallelogramfacetcontaining−vi,alsovl∈∂M(−vi),somi(vl)=1;thisis
3.totradictioncona

Finallythefourlemmaswillbeproved.
Lemma4.3.9.Thereexistexactly34quasi-smoothFanopolytopeshavingan
AS-pointofdegree≤4.Oftheseare10smoothand24singular.
Prhaovingof.ByAS-p[Bat99oints,,b2.5]yand4.2.164.3.3necessarilythereareofexactlydegree104.smoSoletothPFbanoepsingular.olytopesLetleft
vi∈V(P)beanAS-point(asin4.2.17(1))withdeg(vi)=4andvj+vk=via
quasi-primitiverelationforvj,vk∈V(P).Choosealsovl,vrasin4.2.17(2)with
quasi-primitiverelationP.By4.2.16onecandistinguishwithoutrestrictionthe
followingtwocases(correspondingto4.2.17(3)):
CASEI:[vi,vj]and[vi,vk]areedgesofP.
By4.2.8(1)and4.2.13therearewithoutrestrictionthefollowingpossibilities
forPleft(correspondingto4.2.17(2)):
Subcase(a):vl+vr=0.(Piisoftype4a.)InorderforPnottobe
simplicialtherehavetobeatleasttwocontagiouslatticepointsofPiby4.2.13
mandi(vl)4.2.15,=1.soTheletfollowithoutwingtworestrictioncasesformi(vthej)=2=determinationmi(vr).ofBydouble4.2.15(3)pointsthenof
Pi(correspondingto4.2.17(4))arenowonlyleft:
mi(vk)typeofP
8.419.172Subcase(b):vl+vr=vi.(Piisoftype4a.)ForPnotbeingsimplicial
letagainwithoutrestrictionmi(vj)=2andmi(vr)=2.Bysymmetryalso
mi(vk)≥mi(vl).Thefollowingcasesarethenonlyleft:

106

106Chapter4.TerminalGorensteintoricFano3-folds
mi(vk)mi(vl)typeofP
8.31121−viAS-pt:caseI(a)
10.122Subcase(c):vl+vr=−vi.(Piisoftype4a.)By4.2.15(3)mi(vl)=1=
mi(vr).ThenPmustbesimplicial,contradiction.
Subcase(d):vl+vr=vj.(Piisoftype4b.)Bysymmetryletmi(vr)≥
mi(vl).InorderforPtohaveaparallelogramfacet,thefollowingcasesare
4.3.8(1):ybleftonlymi(vj)mi(vk)mi(vr)mi(vl)typeofP
7.812118.812218.1412129.822129.18122210.42222Subcase(e):vl+vr=vj−vi∈V(P).(Piisoftype4b.)Especially
mi(vj)=2.Bysymmetryletmi(vr)≥mi(vl).InorderforPtohavea
parallelogramfacet,thefollowingcasesareonlyleftby4.3.8(2):
mi(vk)mi(vr)mi(vl)typeofP
8.111219.14122Subcase(f):vl+vr=vj+(vj−vi)istheparallelogramrelationofafacet.
(Piisoftype4c.)Especiallymi(vj)=2,andby4.2.15(2)mi(vl)=1=mi(vr).
Sothefollowingtwocasesareleft:
mi(vk)typeofP
7.118.12Subcase(g):vl+vr=vj−viistheparallelogramrelationofafacet.(Pi
isoftype4b.)By4.2.15(1)especiallymi(vj)=mi(vl)=mi(vr)=1.Sothe
followingtwocasesareleft:
mi(vk)typeofP
6.317.142Subcase(h):vl+vr=(vj−vi)−viistheparallelogramrelationofafacet.
(Piisoftype4b.)By4.2.15(1)especiallymi(vj)=2,mi(vl)=mi(vr)=1.So
thefollowingtwocasesareleft:
mi(vk)typeofP
7.712vjAS-pt:caseI(f)

4.3.Classificationofquasi-smoothFanopolytopes107
CASEII:[vi,vj]isanedgeofPandconv(vi,vr,vk,vl)isaparallelogram
facetwithdiagonals[vi,vk]and[vl,vr].(Piisoftype4b.)
Thenallverticesin∂M(vi)aredetermined.Letmi(vr)≥mi(vl).Ifmi(vj)=
2=mi(vk),then−viisanAS-pointofcaseIbecauseof4.3.8(3).Sothe
followingpossibilitiesareleft:
mi(vj)mi(vk)mi(vr)mi(vl)typeofP
6.111117.512118.922112111vj−viAS-pt:caseI
8.712129.322121211vk−viAS-pt:caseI(by4.3.8(3))
8.512219.62221

AS-pLemmaointof4.3.10.minimalTherdeegreexiste5.exactlyTheyar37eallquasi-smosingular.othFanopolytopeshavingan
Proof.Letvi=vj+vkbeanAS-pointasin4.3.9and{vl,vr}thequasi-primitive
collectionwith[vl,vi],[vi,vr]edgesofP.
cases:owtDistinguishCASEI:[vi,vj]and[vi,vk]areedgesofP.
conv(vi,Withoutvr,vr−vrestrictionj,vk)bay4.2.16parallelogramletconv(facetvi,vofj,vPr.)bByeafacet4.2.15(2)of{Pvl,vandr−vj}is
collection.equasi-primitivaalsovl+(Subvrc−asevj)(a)=:vvkl−+vvi.r=So0.by(Piis4.2.15(3)oftypande5a.)4.3.8(1,2,3)Thenvj+there(vrare−vthej)=follovrwingand
left:casesmi(vj)mi(vr)mi(vr−vj)mi(vk)mi(vl)typeofP
111117.12
211118.12
21221111119.118.13
∙1121vk−viAS-pt:deg.4
21221122119.1510.12
112219.4
1222110.5
2222111.1
111229.13
2112210.10
mi(vkSub),caseand(b)mi:(vvjl)+≥vmri(=vlv),i.if(Pmii(isvr)of=typmie(vk5a.)).ItByisvj+symmetry(vr−vletj)m=iv(rvr)and≥
vl+(vr−vj)=vk.By4.3.8(1,3)thefollowingcasesarethenonlyleft:

108Chapter4.TerminalGorensteintoricFano3-folds

mi(vj)mi(vr)mi(vr−vj)mi(vk)mi(vl)typeofP
111117.9
21111vkAS-pt:caseI(a)
211129.5
112118.18
1211∙vr−viAS-pt:deg.4
22111vj−viAS-pt:caseI(a)
2211210.11
122119.10
2221110.7
121219.16
22121vj−viAS-pt:caseI(a)
2212211.4
1222110.9
2222111.2
2222212.1
Subcase(c):vl+vr=−vi.(Piisoftype5a)By4.2.15(3)mi(vl)=1=
mi(vr).Thenvl+(vr−vj)=(vk−vi)−vi.Thereforemi(vk)=2and
mi(vr−vj)=1.Thentherearethefollowingcasesleft:
mi(vj)typeofP
1vk−viAS-pt:deg.4
2vjAS-pt:deg.4
Subcase(d):vl+vr=vj.(Piisstilloftype5a.)Thenvl+(vr−vj)=0.
By4.2.15(3),4.3.8(1,3)thefollowingcasesareleft:
mi(vj)mi(vr)mi(vr−vj)mi(vk)mi(vl)typeofP
111117.4
11211vr−vjAS-pt:deg.4
∙1121vjAS-pt:deg.4
11221vjAS-pt:caseI(b)
211118.15
121118.17
122119.9
12121vk−viAS-pt:caseI(a)
1222110.6
22111vj−viAS-pt:caseI(b)
22211−viAS-pt:caseI(a)
22∙21vjAS-pt:caseI(b)
11112vlAS-pt:deg.4
111229.12
211∙2−viAS-pt:caseI(a/b)
22112again10.6
2212211.3

4.3.Classificationofquasi-smoothFanopolytopes109
Subcase(e):vl+vr=vj−vi∈V(P).(Piisoftype5a.)Especially
mi(vj)=2.Thenvl+(vr−vj)=−vi,soby4.2.15(3)mi(vl)=1=mi(vr−vj).
Soitfollowsby4.3.8(2):
mi(vr)mi(vk)typeofP
11−viAS-pt:caseI(a)
21−viAS-pt:caseI(d)
12−viAS-pt:caseI(b)
10.222Subcase(f):vl+vr=vk.(Piisoftype5b.)Thenby4.2.15(2)mi(vk)=2,
mi(vl)=1=mi(vr−vj).By4.3.8(1,3)itfollows:
mi(vj)mi(vr)typeofP
∙1vk−viAS-pt:deg.4
9.72122vjAS-pt:caseI(e)
Subcase(g):vl+vr=vk−viisaquasi-primitiverelation.Thisisacontra-
4.3.8(4).todictionSubcase(h):vl+vr=vj−viistheparallelogramrelationofafacet.(Piisof
type5a.)Especiallymi(vl)=mi(vj)=mi(vr)=1.Thenvl+(vr−vj)=−vi,
soalsoby4.2.15(3)mi(vr−vj)=1.Sothefollowingtwocasesareleft:
mi(vk)typeofP
7.1012vk−viAS-pt:deg.4
Subcase(i):vl+vr=vj+(vj−vi)istheparallelogramrelationofafacet.
(Piisoftype5b.)By4.2.15(2)especiallymi(vj)=2,mi(vl)=mi(vr)=1.
Thenvl+(vr−vj)=vj−vi,soalsovr−vj∈∂M(−vi)andmi(vr−vj)=1by
ws:folloIt4.2.15(1).mi(vk)typeofP
8.612−viAS-pt:caseI(f)
CASEII:[vi,vj]isanedgeofPandconv(vi,vr,vk,vl)isaparallelogram
facetwithdiagonals[vi,vk]and[vl,vr].(Piisoftype5a.)
By4.2.16letwithoutrestrictionconv(vi,vj,vr−vk,vr)beaparallelogram
facetofP.Thenallverticesin∂M(vi)aredetermined.Bysymmetry(along
[vi,vr])letmi(vj)≥mi(vl),andmi(vr−vk)≥mi(vk),ifmi(vj)=mi(vl).If
mi(vj)=2=mi(vk)ormi(vl)=2=mi(vr−vk),then−viisanAS-pointof
caseIbecauseof4.3.8(3).Soby4.3.8(1,3)thefollowingpossibilitiesareleft:

110Chapter4.TerminalGorensteintoricFano3-folds

mi(vj)mi(vr−vk)mi(vr)mi(vk)mi(vl)typeofP
111117.6
112118.16
12∙11vr−vk−viAS-pt:
Icase1222110.3
21111vj−viAS-pt:deg.4
21211vj−viAS-pt:caseI
22∙11vj−viAS-pt:caseI
21112vj−viAS-pt:caseI
2121210.8

Lemma4.3.11.Thereexistexactly12quasi-smoothFanopolytopeshavingan
AS-pointofminimaldegree6.Theyareallsingular.
Proof.Letvi=vj+vkbeanAS-pointofdegree6withassociatedquasi-
primitiverelation,and{vl,vr}theuniquequasi-primitivecollectionwith[vi,vl]
and[vi,vr]edgesofP.By4.2.16wewilldistinguishthefollowingcases:
CASEI:Thereare3parallelogramfacetsofPcontainingvi.
Letvj,vr−vk,vr,vk,vl=vi+vk−vr,vi−vrbeanorderingof∂M(vi)
accordingto4.2.16(2).(Piisoftype6a.)LetW:={πi(vr−vk),πi(vk),πi(vi−
.)vr}Subcase(a):Thereexistsanelementw∈Wwithwand−wdoublepoints.
Withoutrestrictionletmi(vk)=2andmi(vj)=2.ThenvkisanAS-point,
havingnecessarilydegree≥6,itfollowsmi(vl)=2=mi(vr).Withoutrestric-
tiontherearenowthreecases:
mi(vr−vk)mi(vi−vr)typeofP
12.21113.11214.122Subcase(b):Thereexistsanelementw∈Wwithwdoublepointand−w
simplepoint,butnoelementinWasinsubcase(a).Withoutrestrictionlet
mi(vk)=2andmi(vj)=1.Alsoletmi(vr)≥mi(vl)andmi(vr−vk)≥
mi(vi−vr),ifmi(vr)=mi(vl).By4.3.8(1)therearenowthefollowingcases:
mi(vr−vk)mi(vr)mi(vl)mi(vi−vr)typeofP
9.1111112∙1vr−viAS-pt:deg.5
11.51122Subcase(c):AllelementsinWaresimplepoints,i.e.mi(vr−vk)=1,
mi(vk)=1,mi(vi−vr)=1.Withoutrestrictiontherearethefollowingcases:
mi(vj)mi(vr)mi(vl)typeofP
8.191112∙1vj−viAS-pt:deg.≤5
11.6222

4.3.Classificationofquasi-smoothFanopolytopes111

CASEII:Thereareexactly2parallelogramfacetsofPcontainingviand
thereisnoAS-pointofthetypeasincaseI.
Herethefollowingsublemmaisuseful:
Mi∩SublemmaPisuchthat4.3.12.w+Inwthis=0.situationThenwletisthernotebtheemiddnonzerleopointlatticeofpaointsfacetw,ofwP∈i
withLetthrve,evlattic∈e∂Mp(vioints.)withπi(v)=w,πi(v)=w.Ifadditionallywisa
doublesymmetric,pointv+andv[v=i,vv],awisadiagonalsimpleofappointaralandlelogrv−amvfacisetanofPAS-p,thenoint.visnot
iiProof.Firstthreeremarks:
[vt,v(i)p]aAssumediagonalthereofawereanparallelogramAS-poinfacettvtof∈PV.(PBy)such4.2.16(2)thatvtthen=vvtpw+ouldvqhawithve
degree≤5ordegree6with3parallelogramfacetscontainingvtasincaseI;a
tradiction.con(ii)Letnowwbeadoublepointandassumevweresymmetric.Then
v=vi+(v−vi)wereanAS-point.Soby(i)[vi,v]or[v,v−vi]mustnotbe
diagonals.(iii)Finallyifwisadoublepoint,thenby4.2.15(3)v+v∈{0,vi}.
NowAssumeputwisthesetheobservmiddleationspointoftogether:afacetofPiwiththreelatticepoints.By
be4.2.15(2)symmetric,wisathereforedoublepbyoint(iii)andv[+v,vv−=vvi].aThendiagonal.v−vSoisbyan(ii)AS-pvmoinusttwithnot
ii[[vv,v−i]via,v]adiagonal,diagonal;sobay(ii)convistradictionnottosymmetric,(i).Nowespletweciallybebay(iii)doublevp+voint=andvi
andv−vi∈V(P),i.e.wsimple.
Asanimmediatecorollaryweget:
Sublemma4.3.13.Pidoesnotcontainfacetswiththreelatticepoints,espe-
ciallyPiisoftype6a.
6bProorof.6c.ByBut4.2.13inPitheseconcasestainsPfacetsalsohaswiththreesymmetriclatticepmiddleointspoinonlytsifofPiisfacetsoftwithype
ithreelatticepoints;thisisnotpossiblebytheprevioussublemma.

Therearenowtwocases:
CASEIIA:Thetwoparallelogramfacetscontainingviarecontainedinone
halfspaceoftheplaneRvj+Rvi.
Letalsovrbeelementofthishalfspace.Thenvj,vr−vk,vr,vr−vj,vk,vlis
anorderingof∂M(vi)asin4.2.11(3)andπi(vr)isthemiddlepointonafacet
ofPiwiththreelatticepoints;acontradictiontoSublemma4.3.13.
CASEIIB:Thetwoparallelogramfacetscontainingviarenotcontainedin
onehalfspaceoftheplaneRvj+Rvi.
Withoutrestrictionletconv(vi,vj,vr−vk,vr)beaparallogramfacet.There
arenowtwofurthercasestoconsider:
CaseIIB1:conv(vi,vl,vl−vk,vj)isaparallellogramfacetofP.
Thenvj,vr−vk,vr,vk,vl,vl−vkisanorderingof∂M(vi)asin4.2.11(3).
BySublemma4.3.13wemusthaveπi(vl+vr)=πi(vk).

112Chapter4.TerminalGorensteintoricFano3-folds
Subcase(a):vl+vr=vk.Then(vl−vk)+vr=0,sobySublemma
4.3.12mi(vl−vk)=1.Analogouslymi(vr−vk)=1.Letwithoutrestriction
mi(vr)≤mi(vl).By4.3.8(1)therearethefollowingcases:
mi(vj)mi(vr)mi(vk)mi(vl)typeofP
∙∙1∙vlAS-pt:deg.5
9.2121110.132211∙222vk−viAS-pt:caseIIA
212∙vkAS-pt:deg.≤5
Subcase(b):vl+vr=vk−vi.Then(vl−vk)+(vr−vk)=(vj−vi)−vi
isaquasi-primitiverelation,soby4.2.15(1)mi(vl−vk)=1,mi(vj)=2,
mi(vr−vk)=1andvj−vkisanAS-pointofdegree4.
Subcase(c):vl+vr=(vk−vi)−viistheparallelogramrelationofafacet
ofP.Then(vl−vk)+(vr−vk)=(vj−vi)−2viisaquasi-primitiverelation;
tradiction.conaCaseIIB2:conv(vk,vl−vj,vj,vl,vj)isaparallellogramfacetofP.
Thenvj,vr−vk,vr,vk,vl−vj,vlisanorderingof∂M(vi)asin4.2.11(3).By
Sublemma4.3.13wemusthaveπi(vl+vr)=0.
Subcase(a):vl+vr=0.By4.2.15(3)withoutrestrictionmi(vr)=1.Itis
(vl−vj)+(vr−vk)=−vi,somi(vl−vj)=1=mi(vr−vk)by4.2.15(3).Then
by4.3.8(1,2)thereareonlythefollowingcases:
mi(vj)mi(vk)mi(vl)typeofP
8.2011121∙vj−viAS-pt:deg.5
∙21vk−viAS-pt:deg.5
∙22vk−viAS-pt:caseIIA
Subcase(b):vl+vr=vi.Then(vl−vj)+(vr−vk)=0,somi(vl−vj)=
1=mi(vr−vk)bySublemma4.3.12.Bysymmetrythereareonlythefollowing
consider:tocasesmi(vj)mi(vr)mi(vk)mi(vl)typeofP
8.10111121∙∙vj−viAS-pt:deg.5
22∙1vj−viAS-pt:caseIIB1
12.32222Subcase(c):vl+vr=−vi.Then(vl−vj)+(vr−vk)=−2vi;acontradiction.
Lemma4.3.14.Thereexistnoquasi-smoothFanopolytopeshavinganAS-
pointofminimaldegree≥7.
Proof.Letvi∈V(P)beanAS-pointofdegree≥7.Becauseof4.2.13and
4.2.15(2)thereexistsasymmetricdoublepointw∈Pionthemiddleofafacet
ofPi.Becauseof4.2.15(3)thereexistsanAS-pointv∈V(P)withπi(v)=w.
By4.2.15(2)and4.2.16(2)thereforevisanAS-pointofdegree≤6.

4.4.Tableofquasi-smoothFanopolytopes113
4.4Tableofquasi-smoothFanopolytopes
ThroughoutthesectionletPbeaquasi-smoothFanopolytopewithtoricvariety
X=X(M,ΣP).
Inthetablesbelowallthree-dimensionalGorensteintoricFanovarieties
withterminalsingularitiesarelistedbytheircorrespondingquasi-smoothFano
polytopes.Alsoshownaretheirinterestingnumericalcharacteristics.
Anobviouscombinatorialinvariantofaquasi-smoothFanopolytopePis
thef-vector(f0,f1,f2)givingthenumberofvertices,edgesandfacets.Inthe
followingletalwaysn=f0bethenumberofverticesandpthenumberof
.PoffacetsparallelogramProposition4.4.1.LetPbeaquasi-smoothFanopolytopewithnverticesand
pparallelogramfacets.Then
21f1=3n−6−p,f2=2n−4−p,vol(P)=3n−3.

Proof.LetsbethenumberofsimplicialfacetsofP.Thenthestatementsfollow
fromsolvingthesethreeequationsfors,f1,f2:
f1=3s+4p,f2=s+p,n−f1+f2=2,
2and3.7.5.Alternativelyuse3.7.6.

NextitshallbedescribedhowtodeterminethePicardnumberρXofthese
varietiesbycalculatingthecombinatorialPicardnumberoftheassociatedquasi-
smoothFanopolytopeP,see[Ewa96,V.5].Inthisspecialcasethisleadstothe
wing:folloProposition4.4.2.LetPbeaquasi-smoothFanopolytopewithnvertices
{v1,...,vn}andpparallelogramfacets{F1,...,Fp}.Then
Pic(X)=∼Zn−3−λ(P),ρX=n−3−λ(P),
forλ(P):=rank(M(P)),
whereM(P)isamatrixwithncolumnsandprows,whereeachrowconsists
exactlyoftwoentries1,twoentries−1andtheremainingentries0,suchthat
[vj1,vj2]isadiagonalofFiiffM(P)i,i1=M(P)i,i2=0.
EspeciallythePicardnumberofaGorensteintoricFanovarietywithtermi-
nalsingularitiesdoesonlydependonthecombinatorialstructureofitsassociated
quasi-smoothFanopolytope.
Proof.DefineforafacetFofPtheaffinerelationspace
AR(F):={α∈Qn|αjvj=0,αj=0,andαj=0ifvj∈F}.
vj∈Fvj∈F

114Chapter4.TerminalGorensteintoricFano3-folds

IfFissimplicial,AR(F)=0obviously.IfFisaparallelogramfacetwith
diagonals[vj1,vj3],[vj2,vj4],thenAR(F)=Z(ej1+ej3−ej2−ej4)=Z(ej2+
ej4−ej1−ej3)asiseasilyseenfromthedefinition.
Nowthepropositionfollowsimmediatelyfrom[Eik93,Thm.4.1],which
states:PPic(X)=∼Zn−3−dimfacetsFofPAR(F).

Remark4.4.3.Oneshouldremarkthatinsteadofmechanicallycomputing
therankofthematrixM(P)itisveryoftenpossibletodirectlyderivethe
valueofλ(P)fromthecombinatorialstructureoftheparallelogramfacetsof
thepolytopePbyidentifyingrowswithparallelograms.Thismeansthatcertain
operationsontherowscorrespondtosomeontheparallelograms,startingwith
thesetofparallelogramfacetsofP.Thefollowingguidelines(i)to(iii)give,
whenrecursivelyused,inmostcasesabasisofparallelograms,i.e.thecorre-
spondingrowsinM(P)formabasisoftherowspaceofM(P),therebygiving
).P(λ(i)IfthereisavertexcontainedinonlyoneparallelogramF,calculatea
basisoftherowspacewithoutFandappendFtogetabasisofthewholerow
space.(ii)Ifthereisacycleofparallelograms,i.e.aclosedsequenceofparallelo-
gramssuchthateachpairofadjacentoneshasexactlyonecommonedgeand
anytwoofthesecommonedgeshaveemptyintersection,thenremovingany
parallelogramgivesabasisofthecorrespondingrowspace.
(iii)IfthereareparallelogramsF=conv(a,b,c,d)andF=conv(a,b,c,d)
withthecommonedge[b,c]=[a,d]forb=aandc=d,thentheassociated
rowspaceisalsogeneratedbyFandF:=conv(a,b,c,d).Thishasthead-
vantagethatF,FcanbesubstitutedbyF,F,whereforinstancebiscontained
inoneparallelogramless.ItisimportanttonotethatFisingeneralonlya
latticeparallelogrambutnofacetofPanymore.Butthisisirrelevantforthe
continuingcalculationofλ(P).
Example4.4.4.LetPasusualaquasi-smoothFanopolytope.
1.Ifp=0,ρX=n−3asisexpectedforanonsingulartoricvariety.
2.Ifp≤2,thenλ(P)=p.Ifp=3,thenλ(P)=3exceptwhenthe
parallelogramfacetsofPformaprisma.Inthiscaseλ(P)=2by4.4.3(ii).
3.Thecube8.21hasλ(P)=4.Thisfollowsfromapplying4.4.3(ii)twotimes
togetabasisoffourparallelogramfacets,asisimmediatelycheckedby
4.4.3(i).ThereforeρX=1.Thecubeissimple,sothiscouldhavealso
beenconcludedfrom[Eik93,Thm.4.6].Ascanbeseenfromtheclassi-
fication8.21istheonlyquasi-smoothFanopolytopewithparallelogram
facetsthatissimple,becauseitistheonlyonetosatisfytheequivalent
equationn=2f2−4(see[Eik93,Lemma3.6]).

4.4.Tableofquasi-smoothFanopolytopes

115

Inthetablesbelownowallquasi-smoothFanopolytopesParelistedwith
thefollowinginvariants:n=f0istthenumberofvertices,pisthenumberof
anddegparallelograms(X)=(−facetsKX)3andthef2anthenumticanonicalberofdegreefacetsofofPthe,ρXtheGorensteinPicardtoricnumFbanoer
varietyX=X(M,ΣP)withterminalsingularities,herewehaveby(3.7)
deg(X)=3!vol(P∗).
Firstthelistofall18smoothFanopolytopes(i.e.p=0)inthenotationof
]:[Bat99

n30nf2ρXdeg(X)
64144PB156262
B256256
B356254
B456254
C168352
C268350
C368348
C468348
C568344
D168350
D268346
E1710446
E2710444
E3710442
E4710440
F1812536
F2812536
Nowthelistofall82quasi-smoothFanopolytopeswithp>0,herethe
lastcolumngivesareferencetoProposition4.3.3,iftherearenoAS-points,
orotherwisetotheproofofoneoftheLemmas4.3.9,4.3.10,4.3.11,wherethe
isomorphismtypeaccordingtoProp.4.2.17hasbeendescribed.

n0npf2ρXdeg(X)typeofP
5.15151544.3.3
6.16172544.3.9(II)
6.26172464.3.3
6.36172464.3.9(I(g))
6.46172484.3.3
7.17193484.3.9(I(f))
7.27193404.3.3
7.37193384.3.3
7.47193424.3.10(I(d))
7.57193464.3.9(II)
7.67282464.3.10(II)
7.77193424.3.9(I(h))

116

0n7.87.97.107.117.127.137.148.18.28.38.48.58.68.78.88.98.108.118.128.138.148.158.168.178.188.198.208.219.19.29.39.49.59.69.79.89.99.109.119.129.139.149.159.169.179.18

Chapter4.TerminalGorensteintoricFano3-folds

n7777777888888888888888888888999999999999999999

p1123231131123222223212223346554443333333332222

f29987879119111110910101010109101110101099869910101011111111111111111112121212

ρX3321223434432333332343332221222223333333334444

)X(deg42443840404038363238364240363438363834364038423838383232323030343036383434363230323234363232

type4.3.9(I(d))ofP
4.3.10(I(b))4.3.10(I(h))4.3.34.3.10(I(a))4.3.34.3.9(I(g))))4.3.9(I(f4.3.34.3.9(I(b))4.3.9(I(a))I)4.3.9(I4.3.10(I(i))I)4.3.9(I4.3.9(I(d))I)4.3.9(IIB2(b))4.3.11(I4.3.9(I(e))4.3.10(I(a))4.3.10(I(a))4.3.9(I(d))4.3.10(I(d))I)4.3.10(I4.3.10(I(d))4.3.10(I(b))4.3.11(I(c))IB2(a))4.3.11(I4.3.34.3.11(I(b))IB1(a))4.3.11(II)4.3.9(I4.3.10(I(a))4.3.10(I(b))I)4.3.9(I))4.3.10(I(f4.3.9(I(d))4.3.10(I(d))4.3.10(I(b))4.3.10(I(a))4.3.10(I(d))4.3.10(I(a))4.3.9(I(e))4.3.10(I(a))4.3.10(I(b))4.3.9(I(a))4.3.9(I(d))

4.4.Tableofquasi-smoothFanopolytopes

n0npf2ρXdeg(X)typeofP
10.1104124284.3.9(I(b))
10.2104123304.3.10(I(e))
10.3105112344.3.10(II)
10.4105113284.3.9(I(d))
10.5104123324.3.10(I(a))
10.6104123304.3.10(I(d))
10.7105112304.3.10(I(b))
10.8105113284.3.10(II)
10.9103134344.3.10(I(b))
10.10105112284.3.10(I(a))
10.11104123284.3.10(I(b))
10.12103134304.3.10(I(a))
10.13105112284.3.11(IIB1(a))
11.1116122284.3.10(I(a))
11.2115133284.3.10(I(b))
11.3116122264.3.10(I(d))
11.4115133264.3.10(I(b))
11.5118102264.3.11(I(b))
11.6116122264.3.11(I(c))
12.1128122244.3.10(I(b))
12.2127133244.3.11(I(a))
12.3126144244.3.11(IIB2(b))
13.1139131224.3.11(I(a))
14.11412122204.3.11(I(a))

117

Remark4.4.5.Aswillbeseenfromthelisttheonlyquasi-smoothFanopoly-
top13.1esandwherethealsodualthepairpolar12.3,p14.1.olytopeisThesepquasi-smoolytopesotharearealsothetheself-dualonlyponeolytopthate
14.1satisfyistheexactlyZobstruction3(seedeg3.5.1)(X)and=122.f32is−4henceasgivZ3∗en,inthatis,Prop.the4.4.1.innerpByolytoptheewaony
thethirdpageofthisthesis.Herearethefiguresofthesepolytopes,visualized
bythesoftwarepackagepolymake[GJ00,GJ05]:
Thequasi-smoothFanopolytope12.3(=∼Z3∗):

118

The

TheChapter

othquasi-smo

quasi-smoothFano

anoF4.erminalT

eolytopp

polytope13.1

14.1Gorenstein

(self-dual):

(=∼

Z

3):toric

anoF

3-folds

5Chapter

otsroofsetThe

ductiontroInInthispaperwestudythesetofrootsthatisessentialfordeterminingthe
automorphismgroupofacompletetoricvariety.Usingtheseresultswecangive
criteriaHereforonethesourceofautomorphismmotivationgroupcomesofafromcompletethefollotoricvwingarietyresulttobe(forthereductivdefi-e.
nitionofanEinstein-K¨ahlermetricsee[BS99]or[WZ04]):
Theorem(Matsushima1957).IfanonsingularFanovarietyXadmitsan
Einstein-K¨ahlermetric,thenAut(X)isareductivealgebraicgroup.
In1983FutakiintroducedthesocalledFutakicharacter,whosevanishingis
anotherimportantnecessaryconditionfortheexistenceofanEinstein-K¨ahler
metric.ForanonsingulartoricFanovarietywithreductiveautomorphismgroup
thereisanexplicitcriterion(see[Mab87,Cor.5.5]):
Theorem(Mabuchi1987).LetXbeanonsingulartoricFanovarietywith
Aut(TheX)Freutakiductive.characterofXvanishesifandonlyifthebarycenterofPiszero,
wherePisthereflexivepolytopewithX=∼XP(see3.1.5).
criterionIn[BS99for,theThm.existence1.1]BatofanyrevandEinstein-K¨Selivanoahlervawmetric:ereabletogiveasufficient
Theorem(Batyrev/Selivanova1999).LetXbeanonsingulartoricFano
variety.IfXisLetPbsymmetricethe,ri.e.,eflexivethepgroupolytopofewithlatticeX=∼XP.automorphismsleavingPinvari-
anthasnonon-zerofixpoints,thenXadmitsanEinstein-K¨ahlermetric.
InparticulartheygotasaCorollary[BS99,Cor.1.2]thattheautomorphism
groupofsuchasymmetricXisreductive.Expressedincombinatorialterms
ofthisPisjustcenmeanstrallythatsymmetric.thesetSooftheylatticeaskpedointswhetherinthearelativdirecteproinofteriorsofthisofresultfacets
simpleexists.comIndeedbinatorialthereisproevofena(seeTheoremgeneralization5.3.1(1)toandcompleteProp.toricv5.4.2):arietieswitha
Theorem.LetXbeacompletetoricvariety.
thenIftheAut(grX)oupisofreductive.automorphismsoftheassociatedfanhasnonon-zerofixpoints,

119

120

Chapter5.Thesetofroots

MotivatedbyaboveresultsitwasconjecturedbyBatyrevthatinthecase
ofanonsingulartoricFanovarietyalreadythevanishingofthebarycenterof
theassociatedreflexivepolytopeweresufficientfortheautomorphismgroupto
bereductive.Indeedthereiseventhefollowingmoregeneralresultthathasa
purelyconvex-geometricproof(seeTheorem5.3.1(2i)):
Theorem.LetX∼beaGorensteintoricFanovariety.LetPbethereflexive
polytopewithX=XP.
IfthebarycenterofPiszero,thenAut(X)isreductive.
OnlyveryrecentlyXu-JiaWangandXiaohuaZhucouldprovethatthevan-
ishingoftheFutakicharacterisevensufficientfortheexistenceofanEinstein-
K¨ahlermetricinthetoriccase(see[WZ04,Cor.1.3]):
Theorem(Wang/Zhu2004).LetXbeanonsingulartoricFanovariety.
ThenXadmitsanEinstein-K¨ahlermetricifandonlytheFutakicharacter
vanishes.XofCombinedwiththepreviousresultsthisyieldsageneralizationoftheabove
theoremofMabuchithatisalsoimplicitin[WZ04,Lemma2.2]:
Corollary.LetXbeanonsingulartoricFanovariety.
ThenXadmitsanEinstein-K¨ahlermetricifandonlyifthebarycenterofP
iszero,wherePisthereflexivepolytopewithX=∼XP.
ItisnowconjecturedbyBatyrevthatthisresultmayalsoholdinthesingular
caseofaGorensteintoricFanovariety.
Anothersourceofmotivationthatorginatedthisresearchwastheaimto
givemathematicalexplanationsforobservationsmadebyBatyrev,Kreuzerand
theauthorinthecomputerdatabase[KS04b]of3-and4-dimensionalreflexive
polytopes.Hereoneofthemainresultsisanecessaryconditionfortheauto-
morphismgroupofacompletetoricvarietytobereductivethatisgivenbythe
followingsharpupperboundonthedimension(seeTheorem5.1.25):
Theorem.LetXbead-dimensionalcompletetoricvarietythatisnotaproduct
ofprojectivespaces.
IfAut(X)isreductive,thendimAut(X)≤=d22−2d+4,,forfordd=≥23
Thischapterisorganizedasfollows:
ThefirstsectiondealswiththeautomorphismgroupAut(X)ofad-dimen-
sionalcompletetoricvarietyX.HerethesetofrootsRplaysacrucialpartin
determiningthedimensionandwhetherthegroupisreductive(seeProp.5.1.3).
UsingresultsofCoxin[Cox95]weconstructfamiliesofrootsthatparametrize
thesetofsemisimplerootsS:=R∩−Rinageometricallyconvenientway,
thesearecalledS-rootbases.AsanapplicationweshowinProp.5.1.19thatX
isisomorphictoaproductofprojectivespacesifandonlyiftherearedlinearly
independentsemisimpleroots.WhenAut(X)isreductive,weobtainthebound
dimAut(X)≤d2+2d,withequalityiffX=∼Pd(see5.1.20).Moreoverstudying
thisapproachinmoredetailwegetinProp.5.1.22theexistenceofsomespecial
familiesofrootsthatyieldsseveralrestrictionsonthesetR(see5.1.23and
5.1.24).FromthiswecanderivetheaboveboundondimAut(X)inTheorem
5.1.25.

5.1.Thesetofrootsofacompletetoricvariety

121

Inthesecondsectionwemorecloselyexaminethecaseofad-dimensional
GorensteintoricFanovarietyX=XPassociatedtoareflexivepolytopeP.
HerearootofXisjustalatticepointintherelativeinteriorofafacetofP,
sotheresultsoftheprevioussectionhaveadirectgeometricinterpretation.
ForinstanceweobtainthatPhasatmost2dfacetscontainingrootsofP,with
equalityifandonlyifXistheproductofdprojectivelines(seeCorollary5.2.4).
FurthermoretheintersectionofPwiththespacespannedbyallsemisimpleroots
isareflexivepolytopeassociatedtoaproductofprojectivespaces(seeTheorem
5.2.12).Inthethirdsectionwepresentanddiscussseveralequivalentandsufficient
combinatorialcriteriafortheautomorphismgroupofacompletetoricvariety,
respectivelyaGorensteintoricFanovariety,tobereductive(seeTheorem5.3.1).
Inthefourthsectionwedealwiththeabovementionednotionofasymmetric
toricvariety,andsketchtheproofoftheclassificationofallthree-dimensional
symmetricreflexivepolytopes(seeTheorem5.4.5).
ThefifthsectionisconcernedwithananalogueofthenotionoftheEhrhart
polynomial.Herewedonotcountlatticepointsinmultiplesoflatticepoly-
topesbutwesumthem.Thisyieldsavector-polynomialandwedeterminethe
twoheighestcoefficients(seeProp.5.5.2).Theirvanishingisanotherstrong
sufficientconditionfortheautomorphismgrouptobereductive(seeCor.5.5.5).
Inthelastsectionwecompareallthesecombinatorialconditionsandgive
examples.eralsevTheworkinthelasttwosectionswasdoneincollaborationwithM.Kreuzer.
Summaryofmostimportantnewresultsofthischapter:
∙Acharacterizationofproductsofprojectivespaces(Prop.5.1.19,p.127)
∙Asharpupperboundonthedimensionofthereductiveautomorphism
groupofacompletetoricvariety(Thm.5.1.25,p.129)
∙Ad-dimensionalreflexivepolytopehasatmost2dfacetscontaininglattice
pointsintheirinterior,withequalityifandonlyifisomorphicto[−1,1]d
130)p.5.2.4,(Corollary∙Theintersectionofareflexivepolytopewiththespacespannedbyall
semisimplerootsisareflexivepolytopeassociatedtoaproductofprojec-
tivespaces(Theorem5.2.12,p.133).
∙Wegiveequivalentandsufficientcombinatorialcriteriafortheautomor-
phismgroupofacompletetoricvariety,respectivelyaGorensteintoric
Fanovarietytobereductive;includingthevanishingofthebarycenter
134)p.5.3.1,(Theorem∙Thereareuptoisomorphism31three-dimensionalsymmetricreflexive
polytopes(Theorem5.4.5,p.139).

5.1Thesetofrootsofacompletetoricvariety
Inthissectionthesetofrootsofacompletetoricvarietyisinvestigated,and
someclassificationresultsandboundsonthedimensionoftheautomorphism
ed.hievacaregroup

122

Chapter5.Thesetofroots

ThroughoutthesectionletbeacompletefaninNRwithassociatedcom-
pletetoricvarietyX=X(N,).
Definition5.1.1.LetRbethesetofDemazurerootsof,i.e.,
R:={m∈M|∃τ∈(1):vτ,m=−1and∀τ∈(1)\{τ}:vτ,m≥0}.
Form∈Rwedenotebyηmtheuniqueprimitivegeneratorvτoftheunique
rayτwithvτ,m=−1.ForasubsetA⊆Rwedefineη(A):={ηm:m∈A}.
LetS:=R∩(−R)={m∈R:−m∈R}bethesetofsemisimpleroots
andU:=R\S={m∈R:−m∈R}thesetofunipotentroots.Wesaythat
issemisimple,ifR=S,orequivalentlyU=∅.
FurthermorewedefineS1:={x∈S:ηx∈η(U)}andS2:=S\S1,
analogouslyU1:={x∈U:ηx∈η(S)}andU2:=U\U1.Inparticular
η(S1)∩η(S2)=∅andη(S2)=η(U2).
Usuallytheset−RisdenotedasthesetofDemazureroots(see[Oda88,
nienProp.twhen3.13]),howconsideringeverthenormalsignconfansvofenptionolytopherees.willNoteturnthatoutRtobonlyedepmoreendsconvone-
thesetHereofisraaysdirect(1).combinatoricalproofofawell-knownfact:
Proposition5.1.2.|R|<∞.
Proof.Sinceiscomplete,theoriginisintheinterioroftheconvexhullof
{anvτR}τ∈-basis(1).ofByNRsucSteinitz’shthat0theorem=τ∈1.5.2IkτwvτeforfindpaositivsubseteinI⊆tegers{(1)kτ}conτ∈I.tainingFor
m∈Rthisyields
pos(ηm)=τ∈Ikτvτ,m=kpos(ηm),
ofhenceNR,vτthere,mare∈{−only1,0,..finitely.,kpos(manηmy)}cforhoicesallτfor∈Ithe.coSinceIordinatesconoftainsman∈RRin-basisa
dualR-basisofMR.

ThenForthearoidenottitmy∈RcompweonengettaAut◦(X)one-parameterisasemidirectsubgroupproxmduct:Cofa→Autreductiv(Xe).
algebraicsubgroupcontainingthebigtorus(C∗)dandhavingSasarootsystem
andtheunipotentradicalthat◦isgeneratedby{xm(C):m∈U}.Furthermore
Aut(X)isgeneratedbyAut(X)andthefinitenumberofautomorphismsthat
areinducedbylatticeautomorphismsofthefan.Theseresultsaredueto
Demazure(see[Oda88,p.140])inthenonsingularcompletecase,andweregen-
eralizedconsideredbytheCoxcase[Coofx95a,proCor.jectiv4.7]etoricandvB¨arietuhleryin[B¨uh96[BG99].,BrunsThm.and5.4].InGubpartic-eladze
theularunipthereotenisttheradicalfolloiswingtrivial).result(recallthatanalgebraicgroupisreductive,if
5.1.3.ositionProp1.Aut(X)isreductiveifandonlyifissemisimple.
2.dimAut◦(X)=|R|+d.

5.1.Thesetofrootsofacompletetoricvariety

123

WhenXisnonsingular,itiswell-known(see[Oda88,p.140])thateach
irreduciblecomponentoftherootsystemSisoftypeA.Herewewillgivean
explicitdescriptionofSandη(R)byorthogonalfamiliesofrootsthatwillturn
outtobeusefulforgeometricapplications.
WhenconsideringrootsthereisanalgebraicAnsatzduetoCoxthatwill
bediscussedbelow.Itisveryconvenientforproofshoweverhasverylittle
geometricintuitionattached.Ontheotherhandthereisanapproachdueto
BrunsandGubeladzethatisespeciallyintheGorensteincaseverycloseto
convexgeometry.Weproceedhereinakindofcombinationoftheseideas,
essentiallythedefinitionsareclosetogeometry(andwereintroducedbythe
authorbeforehavingseen[BG02]and[Cox95]),mostresultshoweverareproved
inthesimplestwayusingthepaperofCox.
Definition5.1.4.Apairofrootsv,w∈Riscalledorthogonal,insymbols
v⊥w,ifηv,w=0=ηw,v.Inparticularη−v=ηw=ηv=η−w.
Weremarkthattheterm’orthogonal’maybemisleading,becausemost
standardpropertiesdonothold,e.g.,v⊥wdoesnotnecessarilyimply(−v)⊥w.
Lemma5.1.5.LetB={b1,...,bl}beanon-emptysetofrootssuchthat
ηbi,bj=0for1≤j<i≤l.ThenBisaZ-basisoflin(B)∩M.
Proof.Weprovethebasepropertybyinductiononl.Letx:=λ1b1+∙∙∙+λlbl∈
Mwithλ1,...,λl∈R.Thenλl=−ηbl,x∈Z.Sox−λlbl=λ1b1+∙∙∙+
λl−1bl−1∈M.Nowproceedbyinduction.
Wedefinetwospecialpairwiseorthogonalfamiliesofroots:
Definition5.1.6.LetA⊆R.
ApairwiseorthogonalfamilyB⊆Aiscalled
∙A-facetbasis,ifη(A)={ηb:b∈B}∪{η−b:b∈B,−b∈A}.
∙A-rootbasis,ifA=R∩lin(B).
dimRRemarklin(A)=5.1.7.|B|bWhenyB5.1.5.isIfanA-rofurthermoreotbasis,Bw⊆eSha,vethenlin(A)Prop.=lin(5.1.12B),belohencew
impliesthatA⊆S,AcanbeeasilydescribedbyB,andBisalsoanA-facet
basis.NotehoweverthatingeneralanS-rootbasisisnotafundamentalsystem
fortherootsystemSintheusualsense.
ForarbitraryA⊆RwecannotexpecttheexistenceofanA-rootbasis.
HoweveritisoneofthegoalsofthissectiontoshowthattherearealwaysR-
facetbases(5.1.22(2))andS-rootbases(5.1.17).Toexplicitlyconstructthese
familiesanalgebraicapproachduetoCoxshallnowbediscussed:
In[Cox95]CoxdescribedRasasetoforderedpairsofmonomialsinthe
homogeneouscoordinateringofthetoricvariety.ForthiswedenotebyS:=
C[xτ:τ∈(1)]thehomogeneouscoordinateringofX,i.e.,Sisjusta
polynomialringwhereanymonomialinSisnaturallygradedbytheclass
groupCl(X),i.e.,thedegreeofamonomialτxτkτistheclassoftheWeil
thedivisorrayτ.τkτRecallVτ,thatwhereeacVτhisτ∩theNistorus-ingeneratedvariantbyvprime.divisorcorrespondingto
τ

124

Chapter5.Thesetofroots

WeletYdenotethesetofindeterminates{xτ:τ∈(1)}andMtheset
ofmonomialsinS.Foranyrootm∈Rwedefineτm:=pos(ηm)∈(1)and
xm:=xτm∈Y.Nowthereisthefollowingfundamentalresult[Cox95,Lemma
4.4](withadifferentsignconvention):
Lemma5.1.8(Cox95).Inthisnotationthereisawell-definedbijection
h:R→{(xτ,f)∈Y×M,:xτ=f,deg(xτ)=deg(f)},
m→(xm,xτvτ,m).
τ=τm
Form∈Rwehave
m∈S⇐⇒h(m)∈Y×Y,
inthiscaseh(m)=(xm,x−m).
Thenextresultcanbeusedto’orthogonalize’pairsofroots:
Lemma5.1.9.Letv,w∈R,v=−w,ηv,w>0.
Thenηw,v=0andv+w∈R.
Moreoverp(v,w):=ηv,wv+w∈R,v⊥p(v,w),ηp(v,w)=ηv+w=ηw.
p(v,w)∈Siffv+w∈Siffv∈Sandw∈S.

Proof.Letvcorrespondto(xv,f)andwto(xw,g)asinLemma5.1.8.Itis
xv=xw.Theassumptionimpliesthatxvappearsinthemonomialg.Assume
ηw,v>0.Thenxwwouldappearinthemonomialf.Howeversincev=−w
thisisacontradictiontotheantisymmetryoftheorderrelationdefinedin
[Cox95,Lemma1.3].Theremainingstatementsareeasytosee.

Corollary5.1.10.v∈Uandw∈S1impliesηv,w=0.
Lemma5.1.9definesapartialadditiononRandgeneralizespartsof[BG02,
Prop.3.3]inapaperonpolytopallineargroupsduetoBrunsandGubeladze.
Thesettingthereisthatofsocalled’columnstructures’ofpolytopeswhere
’columnvectors’correspondtoroots.Mostpartsofthislemmawerehowever
alreadyindependentlyknownandprovenbytheauthorasanapplicationof
Corollary5.2.8belowinthecaseofareflexivepolytope.
ForanunambiguousdescriptionofSitisnowconvenienttodefineanequiv-
alencerelationonthesetofsemisimpleroots.
Definition5.1.11.Letv≡w(visequivalenttow),ifv,w∈S,v=wand
η−v=η−w.Inparticularthisyieldsη−v,w=−η−v,−w=1.
Proposition5.1.12.LetA⊆RandB⊆SanA-rootbasispartitionedintot
equivalenceclassesoforderc1,...,ct.Then:
A={±b:b∈B}∪{b−b:b,b∈B,b=b,b≡b}⊆S,
|A|=|B|+it=1ci2≤|B|+|B|2,
η(A)={η±b:b∈B},|η(A)|=|B|+t≤2|B|.

5.1.Thesetofrootsofacompletetoricvariety

125

Proof.Onlythefirstequationhastobeproven:Letm∈A,by5.1.5wehave
mletl=>1.b∈BByλbbfororthogonalitλb∈Zy.weLethalve:=−1≤b∈Bη|,λbm|.=Pro−λceed,bhenceyλinduction≤1foronlall,
bbbb∈B.Assumethereisanelementb∈Bwithλb<0.Lemma5.1.9implies
b+m∈lin(B)∩R=A,sob+m∈Sbyinductionhypothesis.NowLemma
5.1.9yields−m∈A.Henceλb=−1.Thereforeλb∈{1,0,−1}forallb∈B.
Assumel>2.Bypossiblyreplacingmwith−mwecanachievethatthereare
twoelementsb,b∈Bwithλb=1=λb,henceηb=ηm=ηb,acontradiction.
Thereforel=2,andtherearetwoelementsb,b∈Bwithm=b−b.Assume
b≡b.Thennecessarilyη−b,b=0,soηb=ηm=η−b,acontradiction.
Definition5.1.13.ThegradingofthepolynomialringS:=C[xτ:τ∈(1)]
bytheclassgroupCl(X)inducesapartitionofYintoequivalenceclasses:
1.LetY1,...,Ypbetheequivalenceclassesoforderatleasttwosuchthat
thereexistsnomonomialinM\Yofthesamedegree.
2.LetYp+1,...,Yqbetheremainingclassesoforderatleasttwo.
3.LetYq+1,...,Yrbethetheequivalenceclassesoforderonesuchthatthere
existsanmonomialinM\Yofthesamedegree.
4.LetYr+1,...,Ysbetheremainingclassesoforderone.
ByLemma5.1.8orderedpairsofindeterminatescontainedinoneofthe
equivalenceclassesY1,...,YpcorrespondtorootsinS1,orderedpairsinYp+1,
...,YqcorrespondtorootsinS2.Aschangingm↔−mform∈Sjustmeansto
reversethecorrespondingpairofmonomials,weimmediatelyseethat−S1=S1
and−S2=S2.MoreoverLemma5.1.8yieldsthatanyrootinS1isorthogonal,
andnotequivalent,toanyrootinS2.
e:vhaeWp=|η(S1)|,q−p=|η(S2)|=|η(U2)|,r−q=|η(U1)|,r=|η(R)|.
WegetfromLemma5.1.8:
qp|S1|=|Yi|(|Yi|−1),|S2|=|Yi|(|Yi|−1).
i=1i=p+1
Moreoverifwedefinefori=p+1,...,rtheequivalenceclassMiconsisting
ofmonomialsinM\YhavingthesamedegreeasanelementinYi,thenweget:
qr|U1|=|Mi|,|U2|=|Yi||Mi|.
i=q+1i=p+1
Inparticular|U2|=∅implies|U2|≥2.SincebyLemma5.1.8fori=
p+1,...,rnoindeterminateinYicanappearinanmonomialinMi,weobtain
thatv,w∈Uwithηv=ηwanddeg(xv)=deg(xw)areorthogonal.SeeExample
illustration.anfor5.1.15

126

Chapter5.Thesetofroots

dsimplexExampleconv(e5.1.14.1,...,eLet’sd,−elo1ok−∙∙at∙−Xed=),P:whereWee1,let...E,dedisdenoteaZthe-basisdofN.-dimensionalHence
P:=E∗isthereflexivepolytopecorrespondingtod-dimensionalprojective
spaceXd=Pd.ThehomogeneouscoordinateringS=C[x,...,x]istrivially
graded.PAut(X)isreductivewithd2+droots.Fore∗,...,0e∗thendualbasisof
Mthefamilyb1:=e1∗,b2:=e1∗−e2∗,...,bd:=e1∗−1ed∗formsdanS-rootbasis,
whereallelementsaremutuallyequivalent.
Example5.1.15.ForanotherexamplewithX=Xweconsiderthethree-
dimensionalreflexivesimplexP:=conv((1,0,0),(1,3,P0),(1,0,3),(−5,−6,−3))
∗2,with|S|V=(P4.)F={(and−1F,0,0)con,(tain−1,0one,2),an(2,tip−o1,dal−1)pair,(−of1,1,0)}semisimple.Werohavots,edimwhileRSF=
321andF4containtheotherpair.F3,F4eachcontainthreeunipotentroots,pairs
ofdataunipS=otenCt[xro0,xots1,xin2,x3],differenClt(XPfacets)=∼Zare,deg(x1orthogonal.)=degW(xe2)can=1readandthisdeg(offx3)the=
deg2(x4)=22.HenceY1={x1,x2},Y2={x3,x4},p=1,q=r=s=2,
{x1,x1x2,x2}aretheelementsinM\Yofdegree2.XPisjusttheweighted
projectivespacewithweights(1,1,2,2).
Thenextpropositionshowshowtoconstructrootbases:
Propelementiosition∈Ia5.1.16.familyofLetasubsetsubsetKi,jI⊆⊆Yi{of1,.c.ar.,q}dinalitybeci,jgiven.+1Choforose1≤forj≤ianyr.
inDenoteKi,jbywithRi,jthethesamesetfixeofdci,jsecondsemisimpleelement.rootsDefinecorrBesp:=onding∪i∈I,to1≤jor≤iderrRedi,jpandairs
A:=lin(B)∩R.
ThenBisanA-rootbasispartitionedintoequivalenceclasses{Ri,j},and
1any≤rjo≤otiin.AcorrespondsexactlytoanorderedpairinKi,jforsomei∈Iand
rMoreoveranyA-rootbasisisgivenbythisconstruction.
vPr=oof.w,ByhenceBisconstructionanA-roandotbasisLemmawith5.1.8givenηv,wequiv=alence0=ηwclasses.,vforUsingv,w∈LemmaB,
5.1.8andthedescriptionofAinProp.5.1.12theremainingstatementsareeasy
see.to

ForA⊆Sandv∈Awealsoseethat
|{w∈A:ηw=ηv}|=|{w∈A:ηw=η−v}|.
ChoosingI={1,...,q},ir=1foralli,andKi=Yi,weget(seealso
5.1.7):RemarkCorollary5.1.17.S-rootbasesexist,inparticularR∩lin(S)=S.Moreover
qdimRlin(S)=i=1(|Yi|−1).
Remark5.1.18.P:=conv(S)isacentrallysymmetricterminalreflexive
polytopewithV(P)=∂P∩M=S.

5.1.Thesetofrootsofacompletetoricvariety

127

Morepreciselydueto5.1.12thereisanisomorphismoflatticepolytopes
(withrespecttolatticeslin(S)∩MandZc1+∙∙∙+cq)
conv(S)=∼(Zc1⊕∙∙∙⊕Zcq)∗,
whereci:=|Yi|−1fori=1,...,q,andZn=conv(±[0,1]n)isthen-
dimensionalstandardlatticezonotope(see3.5.1).Forastrongerstatement
5.2.12.TheoremseeTheexistenceofanS-rootbasisyields:
Proposition5.1.19.Ad-dimensionalcompletetoricvarietyisisomorphicto
aproductofprojectivespacesifftherearedlinearlyindependentsemisimple
ots.orasecthisInX=∼P|Y1|−1×∙∙∙×P|Yq|−1.
Proof.Letq=1,sothereisanS-rootbasisb1,...,bdwithη−b1=∙∙∙=η−bd.
Assumethereexistsτ∈(1)withτ∈{τb1,...,τbd,τ−b1}.Thenvτ,bi=0
fori=1,...,d,sincebi∈S.Thisimpliesvτ=0,acontradiction.Therefore
(1)isdetermined.Sincenoconeincontainsalinearsubspace,thisalready
impliesX=∼Pd.Thegeneralcaseistreatedsimilarlyandlefttothereader.

AsacorollarywegetfromtheexistenceofanS-rootbasisandProp.5.1.12:
Corollary5.1.20.|S|≤d2+d,withequalityiffX=∼Pd.
MoreoverusingPropositions5.1.12and5.1.16wecannowcharacterizethe
subsetsofSthatadmitrootbases:
Corollary5.1.21.LetA⊆S.Thefollowingconditionsareequivalent:
1.ThereexistsanA-rootbasis
2.A=S∩VforanR-subvectorspaceVofMR
3.R∩lin(x,y)⊆Aforanyx,y∈A
4.A=−A,andifx,y∈Awithx=±yandηx,y>0,thenp(x,y)∈A
Thedetailsoftheproofarelefttothereader(only4.to1.hastobeproven).
Aboveresultsyieldnowthefollowingexistencetheorem:
5.1.22.ositionProp1.ThereexistsanR-linearlyindependentfamilyBofrootsthatcanbepar-
titionedintothreepairwisedisjointsubsetsB1,B2,B3suchthatB1isan
S1-rootbasis,B2isanS2-rootbasis,B1∪B2isanS-rootbasisandB3is
aU1-facetbasissuchthatηb,b=0forallb∈B1∪B2andb∈B3.
HencedimRlin(S)+|η(U1)|=|B|≤d.

128

Chapter5.Thesetofroots

2.ThereexistsanR-facetbasisDthatcanbepartitionedintothreepairwise
disjointsubsetsD1,D2,D3suchthatD1isaU1-facetbasis,D2isa
U2-facetbasis,D1∪D2isaU-facetbasisandD3isanS1-rootbasis.
Hence|η(U1)|+|η(U2)|+dimRlin(S1)=|D|≤d.
Proof.1.Applying5.1.16toY1,...,Yp,respectivelyYp+1,...,Yq,givesthe
existenceofanS1-rootbasisB1,respectivelyanS2-rootbasisB2.Theunion
ofthesetwofamiliesgivesanS-rootbasis.
Letx∈U1.Foranyb∈B1∪B2wehaveηb,x≥0.Ifηb,x≥1,then
by5.1.9wecansubstitutep(b,x)∈U1forxbecauseofηp(b,x)=ηx.Since
theelementsinB1∪B2arepairwiseorthogonal,thisprocessfinallygivesan
elementx∈U1withηb,x=0forallb∈B1∪B2.
Assumenowwealreadyhavex1,...,xl∈U1pairwiseorthogonalsuchthat
anyonesatisfiesthepreviouscondition.Letx∈U1withηx=ηxifori=
1,...,l.Firstwecanassumeasbeforethatηb,x=0forallb∈B1∪B2∪
{x1,...,xl}.Nowallwehavetodoisto”orthogonalize”thesetx1,...,xl,x.
Let’sdothisbyinductiononi=1,...,l:Weassumexj⊥xforj=1,...,i−1.
Sinceηxi,x=0,wesubstitutep(x,xi)forxibecauseofηp(x,xi)=ηxi.Then
x1,...,xlisstillapairwiseorthogonalfamily,andweadditionallygetxj⊥xfor
j=1,...,i.
2.AsbeforeitisnotdifficulttogettheexistenceofaU-facetbasisD1∪D2
suchthatD1isaU1-facetbasisandD2isaU2-facetbasis.LetD3beanS1-root
basis.By5.1.10ηy,x=0forally∈D1∪D2andx∈S1.Foranyelementin
D3wenowjusthavetosuccessivelymodifyD1∪D2inthesamewayasatthe
1.ofend

Corollary5.1.23.1.|η(U)|≤d,whereequalityimpliesthatη(R)=η(U).
2.|η(U)\η(S)|≤codimRlin(S).
3.|η(R)|≤2d,withequalityiffX=∼P1×∙∙∙×P1.
Proof.1.Followsfrom5.1.22(2).2.Followsfrom5.1.22(1).
3.LetDbetheR-facetbasisfrom5.1.22(2),wehave|D|≤d.Bydefinition
η(R)={ηx:x∈D1∪D2}∪{η±x:x∈D3},thisgivestheupperbound.
EqualityimpliesD=D3,i.e.,R=S,withnoelementinDequivalenttoany
other.Applyingthepreviouspropositionwegetthedesiredresult.

WhilethecasewhenMRisspannedbysemisimplerootsiscompletelyclassi-
fied,thereareatleastsomepartialresultsinthecaseofcodimensionone.This
researchwasmotivatedbytheobservationsoftheauthorthattherewereno
semisimplereflexivepolygonswithonlyonepairofroots(seeCorollary5.2.5)
andthatweretherenosemisimplethree-dimensionalreflexivepolytopeswith6
rootsapartfrom[−1,1]3inalistgiventotheauthorbyKreuzer[Kre03a].
Proposition5.1.24.LetdimRlin(S)=d−1.
1.If|(1)|=η(S),then∼ther|eY1|−1existsτ∈|Y(1)q|−\1η(S)suchthat(1)\η(S)⊆
{±τ},andwehaveVτ=P×∙∙∙×P.

5.1.Thesetofrootsofacompletetoricvariety

129

2.Ifq=1,i.e.,|S|=d2−d,then|η(U)|=1andη(S)∩η(U)=∅.
Proof.Letb1,...,bd−1beanS-rootbasis.By5.1.5wecanfindalatticepoint
bd∈Msuchthatb1,...,bdisanZ-basisofM.Lete1,...,eddenotethedual
.Nof-basisZ1.Letτ∈(1)\η(S).Thenvτ,bi=0foralli=1,...,d−1,hence
vτ∈{±ed}.ThesetSisbyconstructioncanonicallythesetofrootsofVτ,so
wecanapplyProp.5.1.19.
2.Letq=1.By5.1.12thisisequivalentto|S|=(d−1)2+d−1=d2−d.
Fori=1,...,d−1thereexistki∈Zsuchthatηi:=ηbi=−ei+kied.There
existsSincekd∈|ηZ(S)suc|h=d,thatthereηd:=ηexists−b1τ=∈e1+(1)∙∙\∙η(+Se),d−w1e+makdyed.assumevτ=ed.Let
x=λ1b1+∙∙∙+λdbd∈M.Wehavex∈Rwithηx=ediffx,ed=−1
andx,ηi≥0fori=1,...,d.Thisisequivalenttoλd=−1,λi≤−kifor
i=1,...,d−1andλ1+∙∙∙+λd−1≥kd.Hencethereexistsarootx∈Rwith
ηx=edifandonlyifk1+∙∙∙+kd≤0.
Ontheotherhandletu:=k1b1+∙∙∙+kd−1bd−1+bd∈M.Thenu⊥
isahyperplanespannedbyη1,...,ηd−1.Wehaveu,ed=1andu,ηd=
k1+∙∙∙+kd.Thereforewhen|(1)|=d+1,wegetu,ηd<0,sothereexists
x∈Rwithηx=ed,necessarilyed∈η(U).Otherwisefor(1)\η(S)={±ed},
theanalogouscomputationfor−edyieldsthateitheredor−edisinη(U).
Assumeη(S)∩η(U)=∅,soS2=∅.UsethefamilyBinProp.5.1.22(1):
SincebyassumptionallelementsinB1∪B2aremutuallyequivalent,however
noelementinS1isequivalenttooneinS2,wehaveB=B2,i.e.,S=S2.This
yields|η(U2)|=d.Since|η(U1)|=1,wegetacontradictionto5.1.23(1).
ForGorensteintoricFanovarietiesthesecondpointcannotsimplybeim-
provedascanbeseenfromExample5.1.15.
ThisresultyieldssharpupperboundsondimAut(X)inthereductivecase:
Theorem5.1.25.LetXbead-dimensional◦completetoricvarietywithreduc-
tiveautomorphismgroup.Letn:=dimAut(X).Then
n≤d2+2d,withequalityonlyinthecaseofprojectivespace.
Ifd=2andXisnotaproductofprojectivespaces,thenn=2.
Ifd≥3andXisnotaproductofprojectivespaces,then
n≤d2−2d+4,
whereequalityholdsiffq=2with|Y1|=2and|Y2|=d−1.
Proof.Letci:=|Yi|−1fori=1,...,q.By5.1.12and5.1.17,wehavel:=
c1+∙∙∙+cq=dimRSand|S|=c12+∙∙∙+cq2+l≤l2+l.Recallfrom5.1.3
thatn=|S|+d.From5.1.19wegetthefirststatementforl=d(orsee
(d−5.1.20).2)2+(dMoreo−v2)er<ford2−the3d+second4.statementwecanassumel=d−1,since
By5.1.24(2)wehaveq>1,sinceissemisimple;inparticulard>2.
Wemayassumec1≤...≤cq.
Ifq=2,thenc1+c2=d−1,henceeitherc1=1andc2=d−2(thisyields
c1c2=d−2),orc1≥2andc2≥(d−1)/2(thisyieldsc1c2≥d−1).

130

Chapter5.Thesetofroots

Ifq≥3,theni<jcicj≥c1(c2+∙∙∙+cq)+c2c3=c1(c1+∙∙∙+cq)+c2c3−
c1c1≥c1(d−1)≥d−1.
Inanycase|S|=c12+∙∙∙cq2+d−1=(c1+...+cq)2+d−1−2i<jcicj≤
(d−1)2+d−1+2(2−d)=d2−3d+4,withequalityonlyforq=2withc1=1
andc2=d−2.
Thefollowingexampleshowsthatthelastboundissharpforanyd≥3:
Example5.1.26.(duetoC.Haase,DukeUniversity)
LetP⊆Rdbethed-dimensionalreflexivepolytopedefinedastheconvexhull
of(2E1∗)×{0}×{1}and{0}×(2Ed∗−2)×{−1},whereEk∗⊆Rkdenotesasin
5.1.14thek-dimensionalreflexivepolytopecorrespondingtoPk.Thisimplies
thatP∩(R1×Rd−2×{0})=∼E1∗×Ed∗−2,NPissemisimplewithdimRS=d−1,
andthelastupperboundintheprevioustheoremisattainedbyXP.

5.2Thesetofrootsofareflexivepolytope
ThroughoutthesectionletPbead-dimensionalreflexivepolytopeinMR.
InthissectionwewillfocusonGorensteintoricFanovarieties,thesevarieties
3.1.5).correspondWhentoPisreflexivereflexivpe,olytopweeshaasvebydescribeddefinitioninthethatthirdthecsethapterofroots(seeRCor.of
thenormalfanNPisexactlythesetoflatticepointsintherelativeinteriorof
.PofetsfacDefinition5.2.1.ThesetRofrootsofPisdefinedasthesetofrootsofN.
Form∈RwedenotebyFmtheuniquefacetofPthatcontainsm,andweagainP
defineηm=ηFmtobetheuniqueprimitiveinnernormalwithηm,Fm=−1.
PForisasubsetsemisimpleA,⊆ifRNitisisconvsemisimple,enienttoi.e.,defineR=F(−RA).:={Fm:m∈A}.Wesay
Ption.MostHereresultsthreeoftheexamplespreviousshallbesectionexplicitlyhavenostatedwa(justdirectusegeometricCorollaryin5.1.23(1),terpreta-
thebasicfact−S1=S1,andCorollary5.1.23(3)):
Corollary5.2.2.ThereareatmostdfacetsofPcontainingunipotentroots.
Corollary5.2.3.IfafacetofPcontainsanunipotentrootandasemisimple
rootx,thenthefacetcontaining−xalsocontainsanunipotentroot.
ifandCorollaryonlyifP5.2.4.∼[−1Ther,1]edareat(isomorphicmost2dasfaclatticetsecpolytopontaininges).roots;equalityholds
=toFgetoracanotherharacterizationexampleweofapplysemisimpleProp.5.1.19reflexiveandpolygonsProp.without5.1.24(2)tousingd=the2
existingdimensionalclassificationterminalF3.4.1.anopTheolytopproeofisareliessmoonoththeFanowpell-knoolytopwne,facte.g.,thata3.1.8(1).two-
lettheCorollaryreflexive5.2.5.pLolytopetePEbebaedefinedtwo-dimensionalasin5.1.14,reflexivei.e.,pX∗olytop=∼e.Pk.Fork∈N>0
Ekk2∗PThenorPP∗isissemisimplesemisimpleiffiffPPisaorPsmo∗isothaFanosmopotholytopFanoeorpPolytop=∼e.E2orP=∼E1.

5.2.Thesetofrootsofareflexivepolytope

131

filledAsansquaresareillustrationunipofotenthesetrootsresultsandweemptgiveythesquareslistofarereflexivesemisimplepolygons,roots:where

4c4b4a3

6b=(6b)*6a=(6a)*5b5a

6d=(6d)*7b=(5b)*7a=(5a)*6c = (6c)*

9=3*8c=(4c)*8b=(4b)*8a=(4a)*Especiallyweseethattype9,whereXPcorrespondstoP2,hasthemaximal
number6ofsemisimpleroots;type8a,i.e.,[−1,1]2,hasthemaximalnumber
4offacetscontainingroots;andthereisnosemisimplepolygonwithprecisely
onepairofroots,asproveninProp.5.1.24(2),respectivelyThm.5.1.25.
Ingeneralthereisanicepropertyofpairwiseorthogonalfamiliesofroots:
Proposition5.2.6.LetBbeanon-emptysetofpairwiseorthogonalroots.
B∈bThenF:=Fbisanon-emptyfaceofPofcodimension|B|,andthe
sumoverallelementsinBisalatticepointintherelativeinteriorofF.
ij=1bjandFi:=∩ji=1Fbj.Orthogonalityimpliesthat{Fb1,...,Fbl}isex-
Proof.LetB={b1,...,bl}with|B|=l.Fori∈{1,...,l}wedefinesi:=
actlythesetoffacetscontainingsl.Thereforesl∈relintFl,andsinceanyl-
codimensionalfaceofPiscontainedinatleastlfacets,wemusthavecodimFl≤
l.Ontheotherhandsi∈Fi+1foralli=1,...,l,soF1∙∙∙Fl,hencewe
obtaincodimFl=l.

ThispropositioncanbeappliedtoaU-facetbasis(see5.1.22(2)):
Corollary5.2.7.IfU=∅,thenF∈F(U)Fisafaceofcodimension|F(U)|≤d.
InparticularifPisnotsemisimple,thenthesumoveralllatticepointsin
thenon-emptyfaceF∈F(U)Fisanon-zerofixpointofAutM(P).

132

Chapter5.Thesetofroots

Tfundamenosharptalenproptheertyresultsofpairsoftheoflatticepreviouspointssectiononwtheebuseoundarytheofelemenataryreflexivbute
padditionolytopeofasrootsdescribined5.1.9in(seeProp.also3.3.1.[BG02This,Def.partial3.2]).additionextendsthepartial
Nowweeasilyget:
LemmaThenv+5.2.8.w∈L∂etPv∩∈MR,andw∈z(∂vP,w∩)M∈Fwith.wMor∈eFovervandw=−v.
vηv,w>0iffz(v,w)=av+wfora≥2.
Inthiscasez(v,w)=(ηv,w+1)v+wandv⊥ηv,wv+w=p(v,w).
ExtendingDefinition5.1.4wemayalsomoregenerallydefineanintrinsic
y:orthogonalitofnotionDefinition5.2.9.v⊥wforv,w∈∂P∩M,ifv+w∈∂Pandz(v,w)=v+w.
Asacorollaryof5.2.6weget:
Corollary5.2.10.Letv,w∈∂P∩Mwithv+w∈∂P.Thenv⊥(z(v,w)−v)
orw⊥(z(v,w)−w).
Moreoverifv,w∈R,thenz(v,w)isintherelativeinterioroftheface
F(v)∩F(w)ofcodimensiontwo.
Remark5.2.11.Ascanbeimmediatelyseenfromtheresultsoftheprevious
forsectiontheset(orofbyrootselemeninthetaryspanobservoftwoations),linearlythereindepareendenessenttiallytsemisimplewoproossibilitiesots.In
thecaseofareflexivepolytopethisshallbeclearlyillustrated:
assumeSoletv⊥v,ww.∈LetSAwith:=vR=∩±linw(.v,wBy).Byorthogonalizing5.1.12{v,w}withisanALemma-root5.1.9basis,weandcan
therearetwocases:
1.v≡w.HenceA={±v,±w,±(v−w)}.
ThepreviouslemmaimpliesforP∩lin(v,w):

v

w2.v≡w.HenceA={±v,±w}.
ThepreviouslemmaimpliesforP∩lin(v,w):

v

wNotethatv≡wifandonlyifv−w∈F(v)∩S.

5.3.Criteriaforareductiveautomorphismgroup133

NowwecanimproveProp.5.1.19bytakingtheambientspaceofsemisimple
rootsintoaccount(recallthedefinitionofEdin5.1.14).
Theorem5.2.12.LetB⊆SbeanA-rootbasisforasubsetA⊆R,and
R1,...,RtthepartitionofBintoequivalenceclassesoforderc1,...,ct.Then
therecar1e+∙∙∙+ctisomorphismsoflatticepolytopes(withrespecttolatticeslin(A)∩M
)ZandttP∩lin(A)=∼P∩lin(Ri)=∼Ec∗i.
=1i=1iInparticulartheintersectionofPwiththespacespannedbyallsemisimpleroots
isagainareflexivepolytopeassociatedtoaproductofprojectivespaces.
Proof.Lett=1,i.e.,allelementsinBaremutuallyequivalent.Letl=|B|≥2,
B={b1,...,bl},b:=b1+∙∙∙+bl.
Claim:P∩lin(b1,...,bl)=conv(b,b−(l+1)bi:i=1,...,l)=∼El∗.
DenotebyQthesimplexontherighthandsideoftheclaim,soQ=∼El∗.
By5.2.6b∈il=1Fbi.Sincebyassumptionη−bi,b=jl=1η−bi,bj=
jl=1η−bj,bj=l,itfollowsfrom5.2.8thatz(−bi,b)=b−(l+1)bi∈F−bi
fori=1,...,l.HenceQ⊆P∩lin(b1,...,bl).Ontheotherhandtheprevious
calculationandorthogonalityalsoimpliesthatQ∩Fb1,...,Q∩Fbl,Q∩F−b1
areexactlythefacetsofthesimplexQ.Thisprovestheclaim.
1.>tLetWedefinefori=1,...,tthereflexivepolytopesQi:=P∩lin(Ri)=∼Ec∗i.
Letxi∈Qi∩Mfori=1,...,t.Weshowbyinductionthatx1+...+xs∈P
fors=1,...,t.Assumes≥2andx:=x1+...+xs−1∈P.Withoutrestriction
0=xs=−x.Assumex∼xs.LetF∈F(P)beafacetcontainingxandxs.
Sincexs∈lin(Rs)∩M,by5.1.5xsisanintegrallinearcombinationofelements
inRs,hencetherehastoexistsomebs∈Rssuchthatb=bsorb=−bssatisfies
ηF,b=−1.ThisimpliesF=F(b).Sincex∈Finthesamewaythereexists
j∈{1,...,s−1}suchthatxj∈F(b).Thisimpliesηb,xj=−1,henceagain,
sincexj∈lin(Rj),by5.1.5thereexistsb∈±Rjsuchthatηb,b=−1,a
contradiction.Henceby3.3.1wegetx+xs∈Pasdesired.
ThereforethepolytopeQ:=it=1P∩lin(Ri)iscontainedinP∩lin(A).
HoweversincethefacetsofQareexactlyQ∩F±bifori=1,...,t,bothpolytopes
havetobeequal.

forPRemark=∼E3∗,5.2.13.whereThethefigureinnerponolytoptheetitleisofthethisconwvexorkhshoullwsofallexactlyrootsthe(isomorphicsituation
toZ3,seeRemark5.1.18)andtheoutersimplexisE3∗.Thevisualizationwas
doneusingtheprogrampolymake[GJ00,GJ05].

5.3Criteriaforareductiveautomorphismgroup
Intoricthisvarietsectiony,respwegivectiveelysevaeralGorensteincriteriafortorictheFanovautomorphismariety,tobgroupeofreductivae.complete

134

Chapter5.Thesetofroots

5.3.1.Theorem1.LetX=X(N,)beacompletetoricvariety.
Thefollowingconditionsareequivalent:
(a)issemisimple,i.e.,Aut(X)isreductive
(b)conv(R)iscentrallysymmetric
(c)conv(R)isacentrallysymmetricterminalreflexivepolytope(with
respecttothelatticeM∩lin(R))withverticesR
(d)x∈Rx=0
(1)∈τ(e)vτ,x=0forallx∈R
(1)∈τIfvτ=0,thenissemisimple.
2.LetXPbeaGorensteintoricFanovarietyforP⊆MRreflexive.
Thefollowingconditionsareequivalent:
(a)Pissemisimple,i.e.,Aut(XP)isreductive
(b)x∈Rx=0
(c)v,x=0forallx∈R
v∈V(P∗)
(d)y∈P∗∩Ny,x=0forallx∈R
(e)bP∗,x=0forallx∈R
(f)rvol(F)=rvol(Fx)forallx∈R,F∈F(P)withηF,x>0
(g)|F∩M|=|Fx∩M|forallx∈R,F∈F(P)withηF,x>0
AnyoneofthefollowingconditionsissufficientforPtobesemisimple:
i.bP=0
ii.m∈P∩Mm=0
iii.bP∗=0
iv.y∈P∗∩Ny=0
v.v∈V(P∗)v=0
vi.AllfacetsofPhavethesamerelativelatticevolume
vii.AllfacetsofPhavethesamenumberoflatticepoints
Conditionvi.impliesv.,e.g.,ifPisasmoothFanopolytope.
Remark5.3.2.Usingthelistofd-dimensionalreflexivepolytopesford≤4and
thecomputerprogramPALPduetoKreuzerandSkarke(see[KS04a,KS04b])
wefoundexamplesshowingthatinthesecondpartofthetheoremthesufficient
conditionsi.−v.arepairwiseindependent,i.e.,ingeneralnoconditionimplies

5.3.Criteriaforareductiveautomorphismgroup135

anyother.Theseexamplescanbefoundinthenextsection.Forinstancethe
polytopePthatsatisfiesbP=0,m∈P∩Mm=0,v∈V(P)v=0,howeverP∗
followingsevencolumnvectorsaretheverticesofafour-dimensionalreflexive
doesnotsatisfyanyofthesethreeconditions.
1000−1−11
01000−10
0010−100
000100−1
Example5.3.3.The”dual”ofconditionv.isnotasufficientcondition:The
followingpolygonisnotsemisimple,howeverthesumofthefiveverticesiszero.

FortheproofofTheorem5.3.1weneedsomelemmas.Thefirstisjusta
ation:observsimpleLemma5.3.4.LetbeacompletefaninNR.
m∈R=⇒vτ,m∈N,
(1)∈τasecthisinm∈S⇐⇒vτ,m=0.
(1)∈τLemma5.3.5.LetbeacompletefaninNR.
LetA⊆Rbeasubsetsuchthat
m∈Akmm=0
forsomepositiveintegers{km}m∈A.ThenA⊆S.
Proof.AssumeA∩U=∅.Thenby5.3.4
0=τ∈(1)vτ,m∈Akmm=m∈A∩Ukmτ∈(1)vτ,m≥1,acontradiction.
Inthecaseofareflexivepolytopethefollowingresultisfundamental:
LLemmaetm∈5.3.6.R.LetDefinePbtheeacdanonical-dimensionalprojerctioneflexivemappalongolytopeminMR.
πm:MR→MR/Rm.
Thenπminducesanisomorphismoflatticepolytopes
Fm→πm(P),
withrespecttothelatticesaff(F)∩MandM/Zm.

136

Chapter5.Thesetofroots

Proof.Prop.3.2.2immediatelyimpliesthatπm:Fm→πm(P)isabijection.
Itisevenanisomorphismoflatticepolytopesby3.1.4(11).
Anotherproofcanbeeasilydoneusingonlythedefinitionofaroot.

Usingthislemmawegetareformulationof5.3.4.NotethatA−B:=
{a−b:a∈A,b∈B}forarbitrarysetsA,B⊆Rd;afacetFofad-dimensional
polytopeQ⊆MRissaidtobeparalleltoRxforsomex∈MR,ifηF,x=0.
Lemma5.3.7.LetPbead-dimensionalreflexivepolytopeinMR.
Form∈RwithF:=Fmwehave:
1.P⊆F−R≥0x,P∩M⊆(F∩M)−Nx,{n∈P∗∩N:n,m<0}={ηm}.
2.P=conv(F,F)iffthereisonlyonefacetFwithηF,m>0.
3.m∈SiffthepreviousconditionissatisfiedandηF,m=1.
InthiscaseF=F−m.FurthermoreFandFarenaturallyisomorphic
aslatticepolytopesand{n∈P∗∩N:n,m=0}={ηm,η−m}.
Lemma5.3.8.LetPbead-dimensionalreflexivepolytopeinMR.
Forv∈V(P∗)wedenotebyv∗∈F(P)thecorrespondingfacetofP.Then
rvol(v∗)v=0.
v∈V(P∗)
Proof.HavingchosenafixedlatticebasisofMwedenotebyvoltheassociated
differential-geometricvolumeinMR=∼Rd.LetF∈F(P)arbitrary.SinceηFis
primitive,itisawell-knownfactthatthedeterminantofthelatticeaff(F)∩M,
i.e.,thevolumeofafundamentalparalleloped,isexactlyηF,henceweget
ofMinkowski(see[BF71,no.60])yieldsF∈F(P)rvol(F)ηF=0.
vol(F)=rvol(F)∙ηF.Theeasydirectionofthesocalledexistencetheorem
Theapproximationapproachinthenextproofisbaseduponanideaof
yrev.BatLemma5.3.9.LetQ⊆MRbead-dimensionalpolytopewithafacetFand
x∈aff(F)suchthatQ⊆F−R≥0x.Forq∈Qwithq=y−λx,wherey∈F
andλ∈R≥0,wedefineA(q):=y−2λx.Thisdefinitionextendsuniquelyto
anaffinemapAofMR.
ThenA(bQ)iseitherintheinteriorofQorintherelativeinteriorofafacet
ofQnotparalleltoRx.Thelastcasehappensexactlyiffthereexistsonlyone
facetF=FnotparalleltoRx.
Proof.FirstassumethereisexactlyonefacetF=FnotparalleltoRx.This
impliesQ=conv(F,F).ChooseanR-basise1,...,edofMRsuchthated=x
andRe1,...,Red−1areparalleltoF.Nowlety∈Fanddefineh(y)∈R≥0
suchthaty−h(y)x∈F.Fork∈N>0letFk(y):=y+∪id=1−1[−1/(2k),1/(2k)]ei
andQk(y):=Fk(y)−[0,h(y)]x.ThenbQk(y)=y−h(y)/2xandA(bQk(y))=
y−h(y)x∈F.LetM:=<e1,...,ed−1>Zandz∈relintF.Foranyk∈N>0

5.3.Criteriaforareductiveautomorphismgroup137

wedefineGk:=(z+M/k)∩FandFk:=∪y∈GkFk(y).Fork→∞the
setsFkconvergeuniformlytoF.ThereforealsoQk:=∪y∈GkQk(y)converges
uniformlytoQfork→∞.ThisimpliesthatbQkconvergestobQfork→∞.
SinceAisaffineandbQkisafiniteconvexcombinationof{bQk(y):y∈Gk}
foranyk∈N>0,alsoA(bQk)isafiniteconvexcombinationof{A(bQk(y):y∈
Gk}⊆Fforanyk∈N>0.ThisimpliesA(bQk)∈Fforanyk∈N>0.Since
AiscontinuousandFisclosed,thisyieldsA(bQ)∈F.Moreoverobviously
A(bQ)∈relintF.
NowlettherebemorethanonefacetdifferentfromFandnotparalleltoRx.
{KThenj}wsucehcanthatchoanoseyaKpjolyhedralsatifiesthesubdivisionassumptionofQofintothefinitelypreviousmanycase.polytopHence,es
sincebQisaproperconvexcombinationof{bKj},alsoA(bQ)isaproperconvex
combinationof{A(bKj)}⊆∂Q.HoweversincenotallA(bKj)arecontainedin
onefacet,A(bQ)isintheinteriorofQ.

PrtoricoofvofarietThey,orfolloemws5.3.1.fromThe5.1.18,first5.3.4partandofthe5.3.5.Sotheorem,letX=whenXPXforisPa⊆MRcompletead-
dimensionalreflexivepolytope,andweconsiderthesecondpartofthetheorem.
(a)and(b)areequivalentby5.3.5.Theequivalencesof(a),(c),(d),(e)and
thesufficiencyofiii.,iv.,v.followfrom5.3.4and5.3.7.
(f)and(g)arenecessaryconditionsforsemisimplicitydueto5.3.7.
Let(f)besatisfiedandx∈R.By5.3.7(1)and5.3.8wehave
rvol(Fx)=rvol(v∗)v,x.
v∈V(P∗),v,x>0
Byassumptionthereisonlyonevertexv∈V(P∗)withv,x>0,furthermore
v,x=1.Hence5.3.7impliesx∈S.
Let(g)besatisfied.Letx∈R,F:=FxandF∈F(P)withηF,x>0.
Dueto5.3.7(1)andbyassumptionthereisabijectivemaph:F→Foflattice
polytopes,i.e.,h(F∩M)=F∩M.Nowthereexistsalatticepointx∈F
withh(x)=x,necessarilyx=−x∈relintF,sox∈S.
Thesufficiencyofvi.,vii.isnowtrivial,5.3.8showsthatvi.impliesv.
Fromnowonletx∈RandAtheaffinemapdefinedasin5.3.9forQ:=P
andF:=Fx.
x−2Letx=i.Abe(0)=satisfied.A(bP)∈ByR,since5.3.7(1)PwiseacancanonicalapplyFanoLemmapolytop5.3.9e.toget−x=
x:=yFinally−kxletforii.kbe∈Nsatisfied.maximal,ForandanyletyT∈F:=∩[yM,x].defineThenxy∈P5.3.7(1)∩Mimplieswith
yyythat−x=A(0)=A|P∩1M|m∈P∩Mm=Ay∈F∩M|P∩M||Ty∩1M|m∈Ty∩Mm
|Ty∩M|
=y∈F∩M|P∩M|A|Ty∩M|m∈Ty∩Mm=y∈F∩M|P∩M|xy.
|Ty∩M|1|Ty∩M|
Hence−xisaproperconvexcombinationof{xy}y∈F∩M,so−x∈R.

138

Chapter5.Thesetofroots

arietiesvtoricSymmetric5.4Definition5.4.1.AtoricvarietyX=X(N,)iscalledsymmetric,ifthe
linearautomorphismgroupofthefanhasonlytheoriginasafixpoint.
AlatticepolytopeQ⊆NRiscalledsymmetric,ifthegroupoflinearauto-
morphismsAutN(Q)ofQhasonlytheoriginasafixpoint.
WhenXisaFanovariety,i.e.,=ΣQforaFanopolytopeQ⊆NR,then
XissymmetriciffQissymmetric.
Inparticularcentrallysymmetriclatticepolytopesaresymmetricandprod-
uctsofprojectivespacesaresymmetric.Moreoversymmetryisinvariantunder
dualizingasobservedbyBatyrev,hencethepreviousdefinitionequalstheone
givenintheintroduction:
Proposition5.4.2.LetP⊆MRareflexivepolytope.Then
PissymmetricifandonlyifP∗issymmetric.
InparticularaGorensteintoricFanovarietyXPissymmetricifandonlyifP
symmetric.isProof.LetG:=AutM(P).WedenotebyFix(G)thesetofelementsinMRthat
arefixedbytheactionofG.SinceGisafinitegroup,thetheoremofMaschke
yieldsanG-invariantR∗-subv∗ectorspaceUofMRsuchthatFix(G)⊕GU=∗MR.
DualizingSymmetryyieldsyieldsFixdim(G)(Fix⊕(GG∗U))==dimNR.(FixHence(G∗)).dimRSo(Fix(GhasG))≤non-trivialdimR(Fixfixp(Goin)).ts
RRiffG∗does.SinceAutN(P∗)=AutM(P)∗,theproofisfinished.

AsexplainedintheintroductionBatyrevandSelivanovaobtainedin[BS99]
fromtheexistenceofanEinstein-K¨ahlermetric,thatifXPisanonsingular
symmetrictoricFanovariety,thenPhastobesemisimple.Theyaskedwhether
adirectproofforthiscombinatorialresultexists.Indeedthereareevenatleast
fiveessentiallydifferentproofs:
Corollary5.4.3.LetP⊆MRbea∗reflexivepolytope.
IfPissymmetric,thenPandParesemisimple.
Proof.Weeasilyseethatthesecondequivalentandeventhefirstfivesufficient
conditionsinthesecondpartofTheorem5.3.1aresatisfied.Foryetanother
proofwecoulduseCorollary5.2.7.

FurthermorethefirstpartofTheorem5.3.1immediatelyyieldsageneral-
izationtocompletetoricvarieties:
Corollary5.4.4.LetXbeacompletetoricvariety.
IfXissymmetric,thenAut(X)isreductive.
SincesymmetrictoricFanovarietieswereofinterestasnaturalexamplesof
theexistenceofanEinstein-K¨ahlermetric,Kreuzergavein[Kre03a]alistof
thesepolytopesuptodimensionfourusingthecomputerdatabase(finding527
polytopesford=4).Theauthorverifiedthelistford≤3:

arietiesvtoricSymmetric5.4.

139

Theorem5.4.5.E2,E2∗,E12,(E12)∗,Z2arepreciselythetwo-dimensionalsym-
metricreflexivepolytopes.Thereare31three-dimensionalsymmetricreflexive
es.olytoppSketchofproof.Ford=2by5.4.3and5.2.5weonlyhavetolookatthereflexive
polytopesE2∗andE12andthefivedelPezzo-surfaces.Nowtheresultisobvious.
Ford=3wecanassumethatPisnotcentrallysymmetric,sinceinthiscase
therealreadyexistsaclassificationduetoWagner[Wag95](forastraightforward
approachbyputtingPinaboxseeProposition6.4.1).
Let’slookatG:=AutM(P)⊆GL(M)=∼GL3(Z).In[Tah71]Tahara
classifiedallfinitesubgroupsofGL3(Z).Fromthislistwegetthatthereare
exactly10non-conjugatedfinitesubgroupsGofGL3(Z)such−id∈GandG
isminimalwithFix(G)={0}.Sixofthesehaveorder4andfourhaveorder6.
Theideaoftheclassificationistodistinguishthese10casesandtousethe
symmetriesinGtodetermineP.
Herewewillshowhowtoproceedintwotypicalsituations:
WeobserveinthelistthatexceptwhenG=∼Z/6Z(whichisaspecialand
−100−100
rathereasycase)thereisanelementg∈GthatisconjugatedbyGL3(Z)to
001010
0−10or001,hencePhasalwaysarotationalsymmetry.
Thesimplestsituationoccurs,whengisexactlythefirstmatrixbefore,in
theZ-basise1,e2,e3.Letv∈V(P)withv3=0.Then(v+vg)/2=(0,0,v3)∈
P∩M.AsPiscanonical,(0,0,v3)∈∂P∩M,andsinceanylatticepoint
ontheboundaryisprimitive,wegetv3=±1.Byassumptionthesumofall
verticesofPiszero,Piscanonicalbutnotcentrallysymmetric,sowecaneasily
provethatP±1:=P∩{x∈MR:x3=±1}areeitherbothone-orbothtwo-
dimensionalfacesofPwith±e3∈P±1.Howeverifbothweretwo-dimensional,
thenbyprojectingalong±e3Prop.3.2.2impliedthatPwerejustaprisma
overcentrallysymmetricfacets,soitselfcentrallysymmetric,acontradiction.
HenceP1=[v,vg]andP−1=[w,wg].Bylookingatthelistwecanassumethat
|G|=4,henceGisabelian.NowsincePissymmetric,itiseasytoseethat
thereexistsh∈Gthatdoesnotfixe3.So(e3)h=(v+vg)h/2=(vh+(vh)g)/2=
(0,0,(vh)3),hencehhastoexchangee3↔−e3.Wesetw:=vh.Letv∼−e3,
thenconv(P−1,v)arecontainedinafacet.Byapplyingg,h,hgwegetthatalso
conv(P−1,vg),conv(P1,w),conv(P1,wg)arecontainedinfacets,soPisjusta
simplex.Thereforewecanassumeby3.3.1thatalsov−e3,vg−e3,w+e3,wg+e3
arecontainedinP0:=P∩{x∈MR:x3=0},andfurthermoreanylattice
pointinP0differentfromthesefourhastobeinst(e3)∩st(−e3).Thisyields
thatπe3(P)=P0=π−e3(P)isacanonical,hencereflexivepolygon,sousing
theclassificationin3.4.1itisastraightforwardcalculationtodetermineP.
010AnothertypicalargumentcanbeexplainedinthecaseG=<σ>=∼Z/4Z
forσ=−100.Letv∈∂P∩M.Hereσ2isconjugatedtothesecond
0−1−1σ2
matrixabove.Wehavem:=v+2v=(0,0,v1+v22+2v3).SincePiscanonical,
impliesthisv1+v2+2v3∈{0,±1,±2}.

140Chapter5.Thesetofroots
Ontheotherhandy:=v+2vσ=(v12+v2,−v12+v2,−2v2)∈P0:=P∩{x∈MR:
(1,1,32),x=0}.Furthermoreif(v1,v2)=(0,0),thenC:=conv(y,yσ,yσ2=
−y,yσ=−yσ)isatwo-dimensionalcentrallysymmetricpolytopecontainedin
P0thatdoesnotcontaininteriorlatticepoints.Nowthelatticepointtheorem
ofMinkowskiimpliesthatthevolumeofC(measuredwithrespecttoaZ-basis
ofP0∩M)isatmost4.Fromthiswecancalculate
v12+v22≤8.
UsingthesetwoequationswecanputPina(capped)cylinder.Nowwehave
atorathercalculatetedioustheG-orbitscalculation,ofthehowlatticetheypcanoinbtseinunitedthistocongettainerareflexivandcehecpkolytopoute.in
Mostothercasesaretreatedinananalogousmanner.

5.5Successivesumsoflatticepoints
Herewecontinuethediscussiononlatticepointsfromsection1.5andsubsection
3.7.1bylookingnotatthenumberoflatticepointsinmultiplesofareflexive
panotherolytopebutconditionattheirforsum.semisimplicitThisleadsy.toTheavmostariantpartofofthethisErhart-psectionwasolynomialdoneandin
collaborationwithV.V.BatyrevandM.Kreuzer.
LetQ⊆MRbeann-dimensionallatticepolytope.
Definition5.5.1.Wedefinefork∈Nthek-thlatticepointsumsQ(k)ofQas
sQ(k):=v∈M.
v∈kP∩M
FurthermoretheweightedbarycenterofQisdefinedas
wbQ:=bQrvol(Q)∈MR.
Thereisnowthefollowingresult,thatmustbeseenintotalsimilarityto
1.5.3:EhrhartoftheoremthePropuniquepositionolynomial5.5.2si(X)∈(Kreuzer,Q[X]suchNill).thatForsi(kany)=isQ∈(k{)1i,..for.,aldl}kther∈Ne.Aexistsnysia
hasdegree≤n+1.
denoteWethedefine(vesQctor-)c:=(os1,.efficient..,sd)of,deandgreletelco∈effNl.(sQThen):=one(coeffhasl(s1),...,coeffl(sd))
coeffn+1(sQ)=wbQ,
1coeffn(sQ)=2F∈F(Q)wbF,
coeff0(sQ)=0.
Thefollowingreciprocitylawholds:
v=(−1)n+1sQ(−k)∀k∈N>0.
v∈relint(kQ)∩M

5.5.Successivesumsoflatticepoints

141

Thisresultshallbeillustratedbytheexampleofthestandardsmoothsim-
plexCn:=conv(0,e1,...,en).Calculationsasexplainedin[Sta86]showthat
111G(X1,...,Xn+1):=Xµ=1−X1Xn+1∙∙∙1−XnXn+11−Xn+1
µ∈pos(Cn×{1})
Thereforeonehase(k)tk=G(1,...,1,t)=1−t1n+1.Thisgives
N∈keCn(k)=k+n
nFixnowi∈{1,...,n}.Nowsi(k)tk=(∂iG)(1,...,1,t)=1−ttn+2.This
N∈kgivesk+n
sCn(k)i=n+1
ObviouslybCn=(1/(n+1))(e1+...+en).Hence
111(wbCn)i=(bCn)irvol(Cn)=n+1n!=(n+1)!=coeffn+1((sCn)i)
Andn112(wbF)i=2(wbconv(e1,...,en))i+(wbconv(0,el:l=1,...,n;l=j))i
1111F∈F(Cn)j=1
=2n!+(n−1)n!=2(n−1)!=coeffn((sCn)i)
Proof.Theproofthatsi(k)givesapolynomialofdegree≤n+1andthe
reciprocitylawfollowsanalogouslyastheproofofthetheoremofEhrhartgiven
µ∈pos(Q×{1})Xµisaspecialrationalfunctionsatisfyingareciprocitylawandthe
inthebookofStanley[Sta86],relyingontheresultthatG(X1,...,Xn+1):=
observationsi(k)tk=(∂iG(X1,...,Xn+1))(1,...,1,t).Thisisexplainedin
N∈kdetailbyKreuzerinthemanuscript[Kre03b].
HoweverthisresultcanalsobederivedfromthegeneralresultThm.1.5.4
duetoBrionandVergne(letφbetheidentitymap).
Forcalculatingthehighestcoefficientwerefineacoveringoftherelative
interiorofQbyn-dimensional1cubesoflength1/k(inthesubspaceaff(Q)∩MR)
centeredatthepointsinQ∩kM.Thisyields
wbQ=detQaffQ=klim→∞wkn=klim→∞Qkkn=coeffn+1(sQ).
xdx1s(k)1
w∈Q∩k1M
Theformulafordeterminingthesecond-highestcoefficientwasdonebythe
author(inspiredbytheproofoftheoriginalErharttheoremduetoBetkeand
]).[BK85KneserItisstraightforwardtoprovethefollowingobservation:

142

Chapter5.Thesetofroots

Transformationformula:LetT:=m→Lm+vbeanaffineunimodular
transformation,soL∈GL(M)andv∈M.Then
sT(Q)(x)=LsQ(x)+eQ(x)vx.
NowwedefineanauxiliaryfunctionwhereQ⊆MRisalatticepolytopeof
:ndimension≤12F∈F(Q)wbF,ifdim(Q)=n
0,ifdim(Q)<n−1
w(Q):=wbQ,ifdim(Q)=n−1
Intheaboveexampleitwasshownthatw(Cj)=coeffn(sCj)forthestan-
dardsmoothsimplexCj:=conv(0,e1,...,ej)forj=1,...,n.Nowtakeany
primitivelatticesimplexCofdimensionj,i.e.,alatticesimplexwherethe
differencesofitsverticesgeneratetheaffinelattice,thenthereexistsanaffine
unimodulartransformationfromCjtoC.Usingtheabovetransformationrule
oneverifysthatalsointhiscasew(C)=coeffn(sC)holds.
Nowthereisawell-knowntheorem[KKMS73]statingthatthereexistsanat-
uralnumberl∈N>0suchthatlQcanbetriangulatedintoprimitivesimplices
.ndimensionofIfQ,Qaretwolatticepolytopesofthedimensionnintersectinginalattice
polytopecontainedinmutualfacets,thenobviouslysQ+Q=sQ+sQ−
sQ∩Q,soalsocoeffn(sQ+Q)=coeffn(sQ)+coeffn(sQ)−coeffn(sQ∩Q).On
theotherhandbylookingatcoeffn+1onegetstheformulawbQ+Q=wbQ+
Usingtheseresultswegetln1/2F∈F(Q)wbF=1/2F∈F(Q)wblF=
wbQ.Thereforeitfollowseasilyw(Q+Q)=w(Q)+w(Q)−w(Q∩Q).
w(lQ)=coeffn(slQ(x))=coeffn(sQ(lx))=lncoeffn(sQ).
Corollary5.5.3(Batyrev,Kreuzer).LetP⊆MRbeareflexivepolytope.
Thenthe(vector-)polynomialsPisdeterminedbythevaluesfork=1,...,d2+1.
overeMor1+d1wbP=d+1wbF,coeffd(sP)=2coeffd+1(sP).
)P(∈FFProof.ByProposition3.7.2weget
sP(k)=v=(−1)d+1sP(−k−1)∀k∈N
v∈relint((k+1)Q)∩M
ThereforesP(k)fork=0,...,d2+1determines≥d+2valuesofsP,hencethe
(vector-)polynomialsP.
Alsoby3.1.4(6)onecanembedanyfacetF∈F(Q)inahyperplaneoflattice
1distanceonefromtheorigin.Thenonecalculatestheformulawbconv(0,F)=
d+1wbF.

Examples5.6.

143

Nowbacktothesetofroots:
FromHerethereciproConditioncity(2b)lawofinProp.Theorem5.5.25.3.1weget:leadsonetoconsidersumsofroots.
FCorollaryoranyi∈5.5.4.{1,.L.et.,dQ}betheranenexistsa-dimensionaluniqueplatticeolynomialpolytoprei(inX)M∈R.Q[X]such
thatri(k)=xiforallk∈N,
x∈R(kQ)
wherFeorRanyisithethesetpofolynomiallatticepri(ointsX)in=(the−1)rnelative(sFinterior)i(−ofX)fachasets.degree≤n.
)Q(∈FFWedefinerQ:=(r1,...,rd).Then
coeffn(rQ)=wbF=2coeffn(sQ).
)Q(∈FF

Thefollowingcorollarygivesasummary(usealso5.3.1,5.5.3):
Corollary5.5.5.LetP⊆MRbeareflexivepolytope.
Thencoeffd(rP)=(d+1)wbP.
Inparticularwegetby5.3.1thefollowingimplications:
1.Pcentrallysymmetric⇒Psymmetric⇒sP=0andrP=0
2.sP=0orrP=0⇒bP=0⇒P,P∗semisimple
3.sP=0⇐⇒sP(k)=0fork=1,...,d2+1
4.sP(1)=0⇒P,P∗semisimple
5.Psemisimple⇐⇒rP(1)=0

Examples5.6Wediscusspossibleimplicationsamongaboveobservedconditionsforareflexive
polytopeP⊆MRbypresentingseveralexamples.
Hereweareinterestedinthefollowingsevenconditions:
symmetricP0:0=s1:P0=b2:P3:thesumofthelatticepointsofPis0(i.e.,sP(1)=0)
4:thesumoftheverticesofPis0
0=r5:P6:Pissemisimple(i.e.,rP(1)=0)

144

Chapter5.Thesetofroots

∗T∗ogether∗withthecorrespondingstatementsforP∗thatwillbeenumerated
as0,1,...,6theseare14conditions.
ByProposition5.4.20holdsiff0∗,andobviously0impliesanyothercon-
dition1,...,6,1∗,...,6∗.Noreverseimplicationholdsascanbeimmediately
seenbybelowtable(e.g.,1doesnotimply4,hencenot0).
Thefollowingtablessummarizepossibleimplications:
⇒123456
1yyyn4cn4ay
2n3byn3bn4bn3by
3n3an3ayn3cn3ay
4n3an3an2yn3an2
5n4byn4bn4byy
6n2n2n2n2n2y
⇒1∗2∗3∗4∗5∗6∗
1n4an4an4an4an4ay
2n3bn4an3bn4an3by
3n3an3an3an3cn3ay
4n3an3an3an2n3ay
5n4bn4bn4cn4bn4by
6n2n2n2n3cn2n2
Notethatbyduality,sincePisreflexive,e.g.,3⇒6∗holdsiff3∗⇒6holds.
Inthetable”y”meansthattheimplicationholds(seeTheorem5.3.1andCorol-
lary5.5.5),however”n3a”forinstancemeansthatexample3abelowgivesa
counterexampleofminimaldimension3.”n2”meansthatonefindsasimple
examplebylookingatthe16reflexivepolygonsinProposition3.4.1.
Inparticularweseethatwehavefoundallpossiblepairwiseimplications
amongtheseconditions.Furthermorenocondition”dualizes”!
MoreoverKreuzer(andtheauthor)observedthat0⇔1⇔2⇔5ford=2,
and0⇔1⇔5ford=3.Foracompletelistofreflexivepolytopessatisfying0,
1or2withd≤4wereferto[Kre03a].
Nextwegivealltheexamples.Theywerefoundusingtheclassificationof
d-dimensionalreflexivepolytopesford≤4andthecomputerprogramPALP
duetoKreuzerandSkarke(see[KS04a,KS04b]).
Herethecolumnvectorsaretheverticesofthereflexivepolytope:
1−10−202
3a00110−2
00001−1
1001−3b011−2
4400−101−201
3c01−2101
00001−1

Examples5.6.

4a4b

4c

100−1−101
0100−100
001−1000
000001−1
−100−1−10000
0100−11−11−1
−1−1−1−1−11100
11100−1−1−1−1
10110−1−1−100
1−1−111−1−1001
111−1−100011
000000−1−111
100110−2
0100−100
001−1000
000001−1

1−001

11−1−0

145

6Chapter

Cenreflexivtrallyepolytopsymmetrices

ductiontroInInreflexivthisecphapterolytopwees;givetheyasacorrespfinalondapplicationtoGorensteinsomeinsightorictinFanocenvtrallyarietiessymmetricwitha
thetoricsetofmorphismrootsofandorderthetwomaximalfixingnonlyumbtheerofneutralverticeselemenandtoflatticetheptorus.ointsHereare
examined.Itisfurtherinvestigated,whethercentrallysymmetricpolytopescan
beembeddedintotheunitlatticecube.Moreoveracompleteclassificationinthe
Thissimplicialgeneralizescaseisgivresultsen,ofevenEwifaldonly[Ewaa88cen]andtrallyhissymmetricstudents[Wpairag95of,facetsWir97].exists.
Summaryofmostimportantnewresultsofthischapter:
∙Twodifferentproofsthatacentrallysymmetricd-dimensionalreflexive
ponlyolytopfores[−1has,1]atdmost(Thm.2d6.1.1,latticep.p148;ointsinProp.interior6.1.4,ofp.149)facets,withequality
∙ProofsymmetricofConjecturesimplereflexiv3.5.2eonptheolytopemaximal(Thm.numb6.2.2,erofp.v149)erticesofacentrally
∙reflexiveCharacterizationpolytopeofwithcasesacenoftrallymaximalnsymmetricumberpairofvoferticesfacetsofa(Thm.simplicial6.2.4,
150)p.∙pairofClassificationfacetsof(Thm.simplicial6.3.1,p.reflexiv151;epCor.olytop6.3.3,esp.with153)acentrallysymmetric
∙Therereflexivearep4,5olytop,15es,20withaisomorphismcentrallyclassessymmetricofpairof2,3,4,5-dimensionalfacets(Thm.simplicial6.3.12,
156)p.∙Anpairydoffacets-dimensionalcanbeemsimplicialbeddedreflexivinto[e−p1,1]olytopd.e(Cor.witha6.4.2,cenp.trally158)symmetric
∙Anyd-dimensionalcentrallysymmetricsimplereflexivepolytopecanbe
embeddedintod/2[−1,1]d.(Cor.6.4.3,p.158)
147

148Chapter6.Centrallysymmetricreflexivepolytopes

∙Aof[−1general,1]dresult(Thm.on6.4.4,embp.edding159;aCor.reflexiv6.4.8,epp.olytop160)eintoasmallmultiple
d∙Alatticedpoin-dimensionalts,withcenequalittrallyyonlysymmetricfor[−1,reflexiv1]dep(Thm.olytop6.5.1,ehasp.at161)most3
∙Apairdof-dimensionalfacetshasatsimplicialmost2d2reflexiv+1eplatticeolytoppeoints,withawithcentrallyuniquenesssymmetricofthe
equalitycase(Thm.6.5.3,p.163)

otsRo6.1Inthissectionthefollowingresultisgoingtobeproved(recallthedefinitionof
thelatticepolytopeE1:=[−1,1]).
Theorem6.1.1.LetP⊆MRbeacentrallysymmetricreflexivepolytopewith
XPthetoricvarietyassociatedtoNP.
|R|1.P=∼E12×Gfora|2R|-codimensionalfaceGofPthatisacentrally
symmetricreflexivepolytope(withrespecttoaff(G)∩Mandaunique
latticepointinrelintG)andhasnorootsitself.
2.AnyfacethasatmostonerootofP.Pcontainsatmost2droots.Hence
dimAut◦(XP)≤3d.
3.Thefollowingstatementsareequivalent:
(a)Phas2droots,i.e.,dimAut◦(XP)=3d
(b)EveryfacetofPcontainsarootofP
(c)P=∼E1d,i.e.,XP=∼P1×∙∙∙×P1
Thefirstpropertyimmediatelyimplies(see5.1.3):
Corollary6.1.2.LetPbeacentrallysymmetricreflexivepolytopewithXP
thetoricvarietyassociatedtoNP.
IfPcontainsnofacetthatiscentrallysymmetricwithrespecttoarootof
P,orthereareatmostd−1facetsofPthatcanbedecomposedasaproduct
oflatticepolytopesE1×F,thenPhasnoroots.
Henceifd≥3andPissimplicial,ord≥4andanyfacetofPissimplicial,
thendimAut◦(XP)=d.
FortheproofofTheorem6.1.1weneedthefollowinglemmathatisaneasy
3.1.4(11):and5.3.7ofcorollaryLemma6.1.3.LetPbeacentrallysymmetricreflexivepolytope.
LetF∈F(P).Then
P=∼E1×FiffFcontainsarootxofP.
InthiscaseFisacentrallysymmetricreflexivepolytope(withrespecttothe
latticeaff(F)∩Mwithoriginx).

erticesV6.2.

149

ProofofTheorem6.1.1.1.Applythepreviouslemmainductively.
2.SinceasjustseenanyfacetofPcontainingarootisreflexive,hencea
canonicalFanopolytope,itcontainsonlyonerootofP.Nowweapply5.1.3(2)
1.and3.SincePasacentrallysymmetricpolytopecontainsatleast2dfacets,we
derivetheequivalencesfrom1.

propAnositionalternativthateusesproofthecanresultsbeofderivtheedfrompreviousthecfollohapter:wingalgebraic-geometric
Proposition6.1.4.LetX=X(N,)beacompletetoricvarietywithcentrally
Then.(1)symmetricdimAut◦(X)≤3d,withequalityifandonlyifX=∼(P1)d.

Proof.By5.1.3and5.1.23(3)itremainstoshowthatforτ∈(1)thereisat
mostonerootm∈Rsuchthatηm=τ.Howevercentralsymmetryyieldsthat
sucSinceharoisotisccomplete,haracterizedthisgivesbythe±vτ,muniqueness=1ofandsuchvaτ,romot.=0forτ=±τ.

erticesV6.2LetP⊆MRbead-dimensionalreflexivepolytope.
ByConjecture3.5.2themaximalnumberofverticesPcanhaveisobtained
onlyforthesimplecentrallysymmetricreflexivepolytope(Z2)2dfordeven.For
d=2adirectproofisgivenin3.3.3,ford=3wehavethefollowingresult:
Proposition6.2.1.Anythree-dimensionalpolytopewithacentrallysymmetric
pairoffacetshasatmost14vertices.
Proof.Bytheclassificationoftwo-dimensionalreflexivepolytopesandLemma
6.1.3wecanassumethattherearenolatticepointsintheinterioroffacets.
Howeveranytwo-dimensionallatticepolytopewithnointeriorpointscanhave
atmostfourverticesasiseasilyseen.SowecanassumetherearetwofacetsF
and−Fwithnormalvectoruhavingatmostfourvertices.Thentheconvexhull
Qofverticesinu⊥haseithernointeriorlatticepoints,soagainthereareonly
atmostfourverticesofQ,orithastheoriginofthelatticeasitsonlyinterior
latticepoint,soQisacanonical,hencereflexivepolygonandhasthereforeby
3.3.3atmostsixvertices.Sooverallthereareatmost4+4+6=14vertices.

Theonlynon-trivialresultvalidinhigherdimensionsisthefollowingtheo-
rem,itwillbeproveninthenextsectiononpage155:
Theorem6.2.2.Conjecture3.5.2holdsfordualsofsimplicialreflexivepoly-
topeshavingacentrallysymmetricpairoffacets.Inparticularitholdsfor
centrallysymmetricsimplereflexivepolytopes.

150Chapter6.Centrallysymmetricreflexivepolytopes

InthesimplicialcasewehaveadirectproofofConjecture3.5.9inthecase
ofacentrallysymmetricpairoffacets.
Forthisweneedthefollowingimportantobservation:
Lemma6.2.3.LetPbeasimplicialreflexivepolytope.LetF∈F(P)such
thatalso−F∈F(P).Wesetu:=ηF∈V(P∗).
LetV(F)={e1,...,ed},ande1∗,...,ed∗bethedualR-basisofNR.For
i=1,...,dwedenotebyFitheuniquefacetofPsuchthatFi∩F=conv(ej:
j=i).
Letv∈V(P)∩u⊥.Wewritev=jd=1qjejasarationallinearcombination.
Thenfori=1,...,d:
qi<0⇐⇒qi=−1⇐⇒v∈Fi.
InparticularthereareI,J⊆{1,...,d}withI∩J=∅and|I|=|J|suchthat
v=ej−ei.
j∈Ji∈I
Moreoverei∗=ηFi−u∈P∗∩Nforanyi∈I.
Proof.AsintheproofofLemma3.5.6wehavefori=1,...,dthatηFi=
u+αiei∗forap∗ositivenaturalnumberαi.Moreoverby3.5.6(1)wealready
knowthatqi=ei,m<0⇔v∈Fi.
Soletqi=ei∗,m<0,v∈Fi.SincethereforeFi∩(−F)=∅,by3.3.1
ηFi−u=αiei∗∈P∗∩N.Hence−1≤αiei∗,−ei=−αi≤−1,soαi=1and
qi=ei∗,m=ηFi−u,m=−1.
Sincev=jd=1(−qj)(−ej)andu=jd=1(−ej∗),applyingtheprevious
statementto−Ffinishestheproof.
Asanimmediateapplicationwedetermineinanydimension,notonlyin
evendimension,thesimplicialreflexivepolytopeswiththemaximalnumberof
verticesinthecaseofacentrallysymmetricpairoffacets(analternativeproof
canbederivedfromtheclassificationresultCor.6.3.3):
Theorem6.2.4.LetP⊆MRbeasimplicialreflexivepolytopethatcontainsa
centrallysymmetricpairoffacets,e.g.,Piscentrallysymmetric.
Then|V(P)|≤3d.Moreover:
|V(P)|=3difandonlyifdisevenandP∗=∼(Z2)2d.
|V(P)|=3d−1ifandonlyifdisoddandP∗=∼[−1,1]×(Z2)d2−1.
Proof.ByTheorem3.5.11weonlyhavetoconsiderthefollowingsituation:
WeLetsetdub:=eoηdd∈andV(|PV(∗P).)|Let=3{dv−1,1....,LetvdF}∈Fdenote(P)thesuchsetthatofvalsoertices−F∈notF(inPF).
FbutinfacetsintersectingFinacodimensiontwoface.
By1Lemmad3.5.6wegetthatV(P)\(V(F)∪V(F))={v∈V(P):u,v=
0}⊆{v,...,v}.Wemayassumethat
{v∈V(P):u,v=0}={v1,...,vd−1}
isasetofcardinalityd−1.WecanenumeratetheverticesofFase1,...,ed
suchthatviisafacetofFi:=conv(vi,ej:j=i)fori=1,...,d−1.Wealso

151theoremClassification6.3.defineforii=1,...,dinthesamewayFi:=conv(wi,ej:j=i)foraunique
.wertexvLetI:={ii∈{1,...,d}:vi∈Fj∀j∈{1,...,d}\{i}}andI:={ii∈
{1,...,d}:wσ(i)∈Fj∀j∈{1,...,d}\{i}}.Leti∈I,then6.2.3yieldsv=
eσ(i)−ei=wforσ(i)∈I.Thisdefinesamapσ:I→I.Moreoverby
symmetrywehav−e1amapψ:I→Idefinedbywi=ei−eψ(i)=vψ(i).
inObvviouslyolution.ψ=Thenσ,theresoσexistisaifixp∈I,oinjtfree:=σp(ierm),k:=utation.σ(j)=Assumei,soσviw=ereejnot−eani
andvj=ek−ej.Hence−ei∼ejandvj∼ej,soLemma3.5.15yieldsa
tradiction.conThereforeσdisjust⊥aproductofdisjointtranspdd−ositions.1Inparticular|I|is
2ev},en.aconAssumetradictionv∈uto.doThendd.weSomavdy∈−assumeF.v=Thereforev,vdhence=I−e=d.{1,..Since.,db−y
3.5.6(3)e1,...,edisaZ-basis,Pisuniquelydetermineduptoisomorphism.

theoremClassification6.3InthecaseofasmoothFanopolytopewherethecentrallysymmetricpairsof
verticesCasagrandespan(seeMR[Cas03bthere]).existsHowaevercompletewecannotexplicitexpectsucclassificationharesultthatforisduegeneralto
centrallysymmetricreflexivepolytopes,sincebytheclassificationofKreuzer
andSkarkethereare150centrallysymmetricreflexivepolytopesalreadyin
dimensionfour.Ford=2wehave3(see3.4.1)andford=3thereare13(see
[Wag95])d-dimensionalcentrallysymmetricreflexivepolytopes.
Howeverin[Wir97,Satz3.3]therewasacharacterizationofcentrallysym-
tices.metricItisreflexivtheepgoalolytopofesthispresensectiontedtothathageneralizevethethis,minimalthenmainumberresultof2willdvber-e
6.3.3.CorollaryintedpresenTheorem6.3.1.LetPbeasimplicialreflexivepolytopewithfacetsF,−F.
trixATherceonsistingexistsaofZ-btheasisverticm1,es..e.1,,m..d.,ofedMofFsuchascthatolumnsinthishasbtheasisfolthelowingma-
02idproperties:A=Cfidd−f∈Matd(N)wheref∈{0,...,d−1}and
C∈Mat(d−f)×f({0,1})suchithatanycolumnofChasanoddnumberof1’s.
Fori=1,...,dweletvdenotetheuniquevertexofPcontainedinafacet
thatintersectsFinthecodimensiontwofaceconv(ej:j=i);obviouslyvi∈ηF⊥
d1iiffvTher=e−ei.existApnyairwisevertexofdisjointPisinsubsets{±e1I,.,....,.,±Ied,⊆v{,1.,....,.v,d}}.andpairwisedis-
l1kjoint=1,...subsets,lsuchJ1,..that.,Jlfor⊆i{∈1,{.1.,..,.d.},d}withweIkhave∩Jk=∅and|Ik|=|Jk|forall
lvi∈ηF⊥⇐⇒i∈Ik⇐⇒vi=ej−eifori∈Ik.
k=1j∈Jki∈Ik
(0In,..this.,0c,1ase,0,.the..i,th0).rowfori∈I1∪∙∙∙Il∪J1∪∙∙∙∪Jlisoftheform
Iffork,k∈{1,...,l}thesetsIkandJkintersect,thenk=k,Ik=Jk
andJk=Ik.

152Chapter6.Centrallysymmetricreflexivepolytopes

Proof.LetV(F)={e1,...,ed},u,Fidefinedasin6.2.3.Considerthefollowing
stepsfortheconstructionofAandm1,...,md:
1.Leti∈{1,...,d}.
Ifvi∈u⊥,by6.2.3ei∗=ηFi−u∈N.
Ifvi∈u⊥,obviouslyvi=−ei,soei∗=ηFi2−u∈21N.
Soinanycasee1∗,...,ed∗∈21N.Inparticularthisyields
2M⊆<e1,...,ed>Z⊆M.
2.WedefineforanarbitraryZ-basisofMthematrixB∈Matd(Z)∩GLd(Q)
consistingofthecolumnse1,...,edinthisbasis.ByTheorem3.6.6thereis
anunimodulartransformationLofMsuchthatA:=LB∈Matd(N)isa
lowertriangularmatrixwithAi,j∈{0,...,Aj,j−1}fori>j.Thenthere
isaZ-basism1,...,mdofMsuchthatthecolumnsofAaree1,...,edin
basis.this3.Thefirstpointyieldsthat2miiscontainedinthecolumnspaceofA,any
diagonalelementofAiscontainedin{1,2}.
4.IfAi,j=1fori>j,thennecessarilyAj,j=2andAi,i=1(againsince
2mjisaZ-linearcombinationofej,...,ed.)
5.Usingthepreviouspointwecaneasilygetbypossiblypermutatingthe
columnsandtherowsthedesiredformofAasablockmatrix(sinceany
vertexofareflexivepolytopeisprimitive,obviouslyf=d).Itremains
toshowthatanycolumnhasanoddnumberof1’s:Bytheprevious
pointwegetsej=2mj+ks=1eik,whereik>j.Weget2ηF,mj=
ηF,ej−k=1eik=−1+s,soshastobeodd.
NowusingCramer’srulewecancalculatethatei∗∈Nifandonlyiftheith
rowofAisoftheform(0,...,0,1,0,...,0).
Bythisobservation,Lemma6.2.3(appliedtoFand−F)andthefirstpoint
intheproofwegetthatitonlyremainstoshowthelastremark:
Soletk,k∈{1,...,l}withIk∩Jk=∅.ByconstructionIk∩Jk=∅,hence
.k=kAssumeIk⊆Jk.Thenthereexistsi∈Ik,i∈Jk.Letj∈Ik∩Jk.Now
wedefineinthedualR-basise1∗,...,ed∗ofNRthevectorw:=ei∗−s=i,jes∗.
ByconstructionitiseasytocheckthatwistheinnernormalofafaceofP
containingasverticesesfors=1,...,dwiths=i,j,aswellas−eiandviand
vj.ThisisacontradictiontoPbeingsimplicial.
HenceIk⊆Jk.InparticularIk∩Jk=∅,soalsoIk⊆Jk.Sincetherefore
|Ik|≤|Jk|=|Ik|≤|Jk|=|Ik|,wehaveIk=JkandIk=Jk.
In[VK85]VoskresenskijandKlyachkocompletelyclassifiedcentrallysym-
metricsmoothFanopolytopes,thiswasextendedbyEwaldin[Ewa88]to
smoothFanopolytopeshavingacentrallysymmetricpairoffacets.Forthis
theydefinedtwospecialclassesofpolytopes:

153theoremClassification6.3.Definition6.3.2.Lete1,...,edbeaZ-basisofM.Letdbeeven.
1.Fd:=conv(±e1,...,±ed,±(e1+∙∙∙+ed))iscalledadelPezzopoly-
tope.ItisacentrallysymmetricsmoothFanopolytope.Thetoricvariety
X(M,ΣFd)isdenotedbyWd,itis(P1)dblown-upintwotorus-invariant
points.WehaveF2=Z2andW2=S3.
2.F˜d:=conv(±e1,...,±ed,−e1−∙∙∙−ed)iscalledapseudodelPezzo
polytope.ItissmoothFanopolytopethatisnotcentrallysymmetric,but
hasacentrallysymmetricpairoffacets.ThetoricvarietyX(M,ΣF˜)is
denotedbyW˜d,itis(P1)dblown-upinonlyonetorus-invariantpoint.d
Nowwegetasacorollaryoftheprevioustheoremafurthergeneralization
oftheclassificationresultsofEwaldandhisstudents:
Corollary6.3.3.LetP⊆MRbeasimplicialreflexivepolytopewithacentrally
symmetricpairoffacets.Then
X(M,ΣP)=∼X(M,ΣP)×(P1)r×Wk1×∙∙∙×Wks×W˜p1×∙∙∙×W˜pt,
wherek1,...,ks,p1,...,ptareeven,rischosentobemaximal,andPisan
n-dimensionalcentrallysymmetricsimplicialreflexivepolytopewith2nvertices
forn:=d−r−k1−∙∙∙−ks−p1−∙∙∙−pt.
Letf:=log2[M:<V(P)>Z].PisasmoothFanopolytopeiffP={0}iff
n=0ifff=0.
LetP={0},i.e.,n≥2,andV(P)={±ei:i=1,...,n}.Thenthere
existsalatticebasisofMsuchthatthematrixAconsistingofthecolumns
e1,...,enisoftheformA=C2idfidn0−fwheref∈{0,...,n−1}and
C∈Mat(n−f)×f({0,1})suchthatanycolumnofChasanoddnumberof1’s
andanyrowofCcontainssome1.
Wecansaysomethingabouttheuniquenessofthisstructuretheorem:
Corollary6.3.4.LetPbeasimplicialreflexivepolytopethathasacentrally
symmetricpairoffacets.
Theninanyproductrepresentationasgiveninthepreviouscorollarythe
numbersk1,...,ks,p1,...,ptareuniquelydeterminedbythecombinatorialtype
ofthepolytopeP.Thenumbersf,nandrareuniquelydeterminedbythe
isomorphismclassofthelatticepolytopeP.
MoreoverthematrixAisindependent(uptomultiplicationbyamatrixin
GLn(Z)andpermutationofcolumns)ofthechosenpairoffacets.
InparticularanytwofacetsofPareisomorphicaslatticepolytopes,i.e.,
thercialelyisanyatwolatticefacetsofautomorphismPhaveoftheMsamemappingnumberoneoffaclatticetepontooints.theother.Espe-
Proof.Firstlookingattheprimitivecollections(seesection4.1)ofFkiandF˜pj
asdescribedin[Cas03b]itiseasytoseethatthenumbersk1,...,ks,p1,...,pt
arecombinatorialinvariants.
NowletF1,F2∈F(P)withF1=F2;theydefinetwoisomorphicproduct
representationsofP,wherewedenoteP1andP2fortherespectivepolytopes
definingX(M,ΣP)×(P1)r,whereitisnotaprioriobviousthatristhesame

154Chapter6.Centrallysymmetricreflexivepolytopes

forP1andP2.Againbylookingattheprimitivecollectionsweseethatnec-
essarilyP1andP2havetobeisomorphic.LetC1bethematrixconsistingof
thecolumnsofP1.ThefacetofP1correspondingtothe(restrictionofthe)
pfacetermF2utatinghasastheverticescolumnswjustegetplusaorlowerminustriangularthecolumnsmatrix.ofNoC1w.weAftermopdifyossiblythe
latticebasisbyexchangingeiwith−ei,ifthecorrespondingcolumnhas−2on
thediagonal;thenwejusthavetoaddtheithrowtoanyrowthathas−1as
anentryinithcolumn.Nowtherehastobeaunimodularmatrixsuchthatthe
multiplicationwiththismatrixyieldsthematrixC2.Finallyitiseasytosee
thatbymultiplicatinga”normalformmatrix”C1thenumberofrowshaving
onlyone1canatmostdecrease;especiallyitisleftinvariant,iftheresultis
againanormalformmatrix,ashereC2.Hencethenumberr(andtherefore
alson)isaninvariantofP.

FRemarkurthermore6.3.5.onecanHencecheckCorollarythatany6.3.3ofgivtheseesawmatricesell-definedindeedmatrixdefinesanormalcenform.trally
symmetricsimplicialreflexivepolytopewiththeminimalnumberofvertices,
thisandisthereisexaminedanandexplicitdiscussedcriterioninwhen[Wir97two,ofSatzthese3.3,Satznormal3.9].formsareequivalent,
WegettheoriginalresultofEwaldin[Ewa88]undermilderassumptions:
verticCorollaryesspanthe6.3.6.latticLetePM,⊆andMRbeassumeathatsimplicialPhasraeflexivecentralplyolytopesymmetricwhereptheair
ets.facofThenthecorrespondingtoricvarietyX(M,ΣP)isjustaproductofprojective
lines,delPezzovarietiesandpseudodelPezzovarieties,inparticularitis
nonsingular.Wealsogetthefollowingresult(wejusthavetoapply[Oda88,Cor.1.16]):
sociateCorollarydfanhas6.3.7.aAcnyentrQally-factorialsymmetricGorpairensteinofdtoricF-dimensionalanovarietyconeswherisethethepras-o-
jectionforthequotientofaproductofprojectivelines,delPezzovarietiesand
pseudodelfPezzovarietieswithrespecttotheactionofafinitegroupisomorphic
to(Z/2Z)forf≤d−1.
Thecombinatorialstatementsoundsrathersurprising:
metricCorollarypairof6.3.8.facetsAnyiscsimplicialombinatorialrlyeflexivepisomorphicolytopetothatasmohasothacFentranoalplyolytopsym-e
havingacentrallysymmetricpairoffacets.
Oneshouldnotbemisledbythisresult:Withoutthesymmetryassumption
thecatedcomthanthebinatoricsoneofofsmosimplicialothFanoreflexivpeolytoppes.olytopescanbemuchmorecompli-
NowwearegoingtoproveTheorem6.2.2.Forthisweneedthefollowing
remark:

theoremClassification6.3.

155

(seeRemarkalso[VK856.3.9.]Byorsetting[Cas03be])0:=that−e1the−∙∙∙−facetseditofisFahastraighveastforwvarderticescalculationprecisely
d{±ej:j=0,...,d,j=i}forfixedi∈{0,...,d}whereexactlyhalfofthe
signsareequalto+1andtheothersareequalto−1.Henceweget:
|F(Fd)|=(d+1)dd.
2Injustthesameway(see[Cas03b])wecancalculate
|F(F˜d)|=ddd−1+d.
d
2i=2di
ProofofTheorem6.2.2.ByCorollary6.3.3wecanassumethatthesimplere-
flexivepolytopeP⊆MRiscombinatoriallyisomorphicto[−1,1]l×(Fk1)∗×
∙∙∙×(Fks)∗×(F˜p1)∗×∙∙∙×(F˜pt)∗,wherel+k1+∙∙∙+ks+p1+∙∙∙+pt=d
fork1,...,ks,p1,...,pteven.Nowastandardinductionargumentshowsthat
2n
2n2n−1+2n<(2n+1)2n≤6n
ni=nin
forn∈N≥1,withequalityattherightonlyforn=1.Hencetheprevi-
ousremarkandthefactthatalso|V([−1,1])|=2<61/2yields|V(P)|≤
6l/26k1/2∙∙∙6ks/26p1/2∙∙∙6pt/2=6d/2,whereequalityimpliesthatdiseven
andPiscombinatoriallyisomorphictoZ2d/2.InthiscasehoweverP∗isa
simplicialreflexivepolytopewith3dverticesandacentrallysymmetricpairof
facets,henceP∗=∼(Z2d/2)∗by6.2.4(oragaindirectlybyCor.6.3.3andCor.
6.3.4).

Thereisonespecialcentrallysymmetricreflexivepolytopethatisextreme
withrespecttothenumberoflatticepoints(seeTheorem6.5.3):
Definition6.3.10.WedefineforaZ-basism1,...,mdofMthereflexivepoly-
topeDd:=conv(±(2m1+md),...,±(2md−1+md),±md).Thetoricvariety
X(M,ΣDd)isdenotedbyDd.
ThepolytopesDdhavethefollowingbasicpropertiesthatarestraightfor-
erify:vtoardwRemark6.3.11.Ddisacentrallysymmetricsimplicialreflexivepolytope.The
dualpolytopeDd∗={md∗−x:x∈id=1−1{0,1}mi∗}isaterminalreflexive
polytope(inthedualZ-basisofm1,...,md).
IfacentrallysymmetricsimplicialreflexivepolytopePisgivensuchthat
2idd−10
A=1∙∙∙11,
inthenotationofTheorem6.3.1,thenP=∼Dd.
TheremainderofthissectionwillapplyCor.6.3.3tolowerdimensions.
Herewewillprove:

156Chapter6.Centrallysymmetricreflexivepolytopes

Theorem6.3.12.Ford=2,3,4,5thereareexactly3,4,10,14isomorphism
classesofd-dimensionalcentrallysymmetricsimplicialreflexivepolytopes,and
4,5,15,20isomorphismclassesofd-dimensionalsimplicialreflexivepolytopes
withacentrallysymmetricpairoffacets.
Arigorousproofofthepreviousresultwasinthecentrallysymmetriccase
uptonowonlyknownford≤3.Ford=4thecentrallysymmetriccaseof
upto10verticeswasdealtwithbyrathercomplicatedandlongcalculationsin
5.11].Satz,[Wir97Theproofissplitinfourparts(herewealwaysdenoteX=X(M,ΣP)).It
isimportanttonotethatbyCor.6.3.3weactuallyonlyhavetodealwiththe
minimalcaseof2nvertices!
Example6.3.13.Weclassifyalltwo-dimensionalreflexivepolytopesPwitha
centrallysymmetricpairoffacets,Pisnecessarilysimplicial.
ByCor.6.3.3wehaveX=∼W˜2orX=∼W2orX=∼P1×P1,ifn=0.
Finallyletn=2andA=1210.By6.3.11wehaveP=∼D2=∼E12,i.e.,
X=∼D2.
Example6.3.14.Weclassifyallthree-dimensionalsimplicialreflexivepoly-
topesPwithacentrallysymmetricpairoffacets.
ByCor.6.3.3wehaveX=∼P1×W˜2orX=∼P1×W2orX=∼P1×P1×P1,
002ifn=0.Ifn=2,thenX=∼D2×P1by6.3.13.
Finallyletn=3.ThenA=020.By6.3.11wehaveX=∼D3.
111Example6.3.15.Weclassifyallfour-dimensionalsimplicialreflexivepolytopes
Pwithacentrallysymmetricpairoffacets.
ByCor.6.3.3wehaveX=∼W˜4orX=∼W4orX=∼(P1)2×W˜2orX=∼
W2×W˜2orX=∼W˜2×W˜2orX=∼(P1)2×W2orX=∼W2×W2orX=∼(P1)4,
ifn=0.Ifn=2,thenX=∼D2×(P1)2orX∼=D2×W2orX=∼D2×W˜2by
6.3.13.Ifn=3,thenX∼=D3×P1by6.3.14.
Finallyletn=4andA=A,inparticular|V(P)|=8.Heretherearethree
(non-unimodularlyequivalent)possibilitiescorrespondingtof=1,2,3:
2000
1.A=11011000.WedenotethetoricvarietyXbyX4.
10012000
2.A=10021000.WehaveX=∼D2×D2.
10102000
3.A=00022000.By6.3.11wehaveX=∼D4.
1111

6.4.Embeddingtheorems157
Example6.3.16.Weclassifyallfive-dimensionalcentrallysymmetricsimpli-
cialreflexivepolytopesPwithacentrallysymmetricpairoffacets.
ByCor.6.3.3wehaveX=∼P1×W˜4orX=∼P1×W4orX=∼(P1)3×W˜2or
X=∼P1×W2×W˜2orX=∼P1×W˜2×W˜2orX=∼(P1)3×W2orX=∼P1×W2×W2
orX=∼(P1)5,ifn=0.Ifn=2,thenX=∼D2×(P1)3orX=∼D2×P1×W2or
X=∼D2×P1×W˜2by6.3.13.Ifn=3,thenX=∼D3×1(P1)2orX=∼D3×W˜12
orX=∼D3×W2by6.3.14.Ifn=4,thenX=∼X4×PorX=∼D2×D2×P
orX=∼D4×P1by6.3.15.
Finallyletn=5andA=A,inparticular|V(P)|=10.Herethereare
three(non-unimodularlyequivalent)possibilitiescorrespondingtof=2,3,4
(f=1isnotpossibledueto6.3.3,sincetherewouldbeanevennumberof1s
column):firstthein20000
02000
1.A=100.UptopermutationofrowsandcolumnsChas
C010
thefollowing0form:01
0111(a)C=10.WedenotethetoricvarietyXbyX5.
1111(b)C=11.Aswasobservedin[Wir97,Bemerkung3.8]this
matrixisequivalenttothepreviousone,soX=∼X5.
20000
02000
2.A=00200.WehaveX=∼D2×D3.
1011010010
20000
02000
3.A=00200.By6.3.11wehaveX=∼D5.
1111100020
theoremseddingbEm6.4WedareinterestedinfindinganembeddingofareflexivepolytopeP⊆MRinto
kE1fork∈Nsmall,i.e.,infindingalatticeautomorphismofMthatmaps
Pinalatticepolytopeisomorphicto[−k,k]d,ormoresimplyput,infindinga
latticebasisofMsuchthatanyvertexofPhascoordinatesin[−k,k].
ThereisaconjectureduetoEwald(see[Ewa88])thatanyd-dimensional
smoothFanopolytopecanbeembeddedintheunitlatticecube[−1,1]d.Itis
provenford≤4bytheclassificationorunderadditionalsymmetries.Itis
wrongforsimplicialreflexivepolytopes,e.g.,type9inProp.3.4.1contains10
latticepoints.Howeverwestillhavethefollowingwell-knownresult:

158Chapter6.Centrallysymmetricreflexivepolytopes

Proposition6.4.1.Fordd≤3wecanalwaysembeddacentrallysymmetric
reflexivepolytopePintoE1.
Proof.Forthiswechooseby3.1.8(1,2)aZ-basisb1,...,bdoflatticepointsin
∂P∗,soP⊆{λibi∗:λi∈{−1,0,1}}forthedualZ-basisb1∗,...,bd∗.
Thereisnosuchresultford≥4.ForinstanceletPbethe4-dimensional
centrallysymmetricreflexivepolytopePinExample3.2.6(thisisalsoX4in
6.3.15).AssumeP∗couldbeembeddedasalatticepolytopein[−1,1]4.Then
([−1,1]4)∗wouldbealatticesubpolytopeofPwiththesamenumberofvertices.
SincePisterminal,thiswouldbeanequality,acontradiction.Thisexample
istakenfrom[Wir97,Kapitel4]wherethistopicisthoroughlydicussed.There
itisshownin[Wir97,Satz4.4]thatcentrallysymmetricpolytopeswiththe
minimalnumberofverticescanalwaysbeembeddedintheunitlatticecube.
UsingTheorem6.3.1wecanproveageneralization:
Corollary6.4.2.LetPbeasimplicialreflexivepolytopewithacentrallysym-
metricpairoffacets.ThenPcanbeembeddedintoE1d.
Proof.Sincethe(pseudo)delPezzopolytopesarebydefinitioncontainedinE1d,
byCorollary6.3.3wejusthavetoshowthatwegetbyrowoperationsonAa
matrixcontainingonly{−1,0,1}.ForthisweassumeAj,j=2,thenthereisa
i>j(minimal)suchthatAi,j=1.Nowwesubtracttheithrowfromthejth.
Weproceedbyinductiononj.

Duetothestructuretheoremofthelastsectionwecanalsoprovethefol-
lowingresult(notethatby6.4.1andthepreviousexamplethisboundissharp
4):dfor≤Corollary6.4.3.LetPbeasimplicialreflexivepolytopewithacentrallysym-
metricpairoffacets.ThenP∗canbeembeddedinto2dE1d.
Proof.Sincedthedualsofthe(pseudo)delPezzopolytopesarealwayscon-
tainedinE1(forthisuse6.3.9and[Cas03b]),byCor.6.3.3wecanas-
02idfsumethatP=conv(±e1,...,±ed)⊆MR.Letm1,...,mdbetheZ-basis
02id/1ofMinTheorem6.3.1suchthatA=Cidd−f∈Matd(N).Then
fA−1=−C/2idd−f.Nowtherowsarepreciselythecoordinatesofthe
dualR-basise1∗,...,ed∗(inthedualZ-basism1∗,...,md∗).Furthermoreforany
facetF∈F(P)wehaveηF=±e1∗±∙∙∙±ed∗∈Nforsomesigns±.Hence
theverticesofP∗havecoordinatesin[−d/2,d/2]withrespecttothelattice
basism1∗,...,md∗.
Themainresultofthissectionisthefollowingcoarsebutmoregeneral
theorem:eddingbemTheorem6.4.4.LetPbeareflexivepolytopeofdimensiond≥3withafacet
F∈F(P)havingavertexthatisonlycontainedin(d−1)facetsofF(e.g.,F
setWesimple).isw:=lcm{u,v+1:u∈V(P∗),v∈V(P)}.

theoremseddingbEm6.4.

159

ThenPcanbeembeddedincE1d,wherec∈O(dw2)isaconstantthatdepends
onlyondandw.
Moreprecisely:cisapositivenaturalnumberwith
c≤min{w((d−2)(w−1)+z),2w((d−1)w−1)},
wherezdenotesthegreatestproperdivisorofw.
Proof.Leted∈V(F)havetheassumedproperty.Thereexiste1,...,ed−1∈
V(F)suchthatfori=1,...,d−1thesetconv(ej:j=i)iscontainedina
facetHiofF.Fori=1,...,d−1wedefineFi∈F(P)suchthatFi∩F=Hi.
AgainasintheproofofLemma3.5.6wehavefori=1,...,d−1that
ηFi=u+αi∗ei∗forηFa−upositivNenaturalnumberdαi;∗moreoverαi=ηF∗i,eiN+1,
thisimpliesei=αii∈w.Sinceu=−i=1eithisimpliesalsoed∈w.In
particularwM⊆<e1,...,ed>Z⊆M.
WedefineMu:=M∩u⊥,andforafixedZ-basism1,...,md−1ofMuthe
matrixB∈Matd−1(Z)∩GLd−1(Q)consistingofthecolumnse1−ed,...,ed−1−
ed.ByTheorem3.6.6thereisanunimodulartransformationLofMusuchthat
A:=LB∈Matd−1(N)isalowertriangularmatrixwithAi,j∈{0,...,Aj,j−
1}fori>j.
MoreoversincewMuiscontainedinthecolumnspaceofA,anydiagonalel-
ementofAisapositivedivisorofw.NowwedefinethematrixA∈Matd(Z)as
theblockmatrixwhereintheupperleftAisputandthedthrowhasonesevery-
where.SinceM=Mu⊕ZedobviouslyintheZ-basism1,...,md−1,md=edof
Mthecolumnsa1,...,adofAaretheimagesofe1,...,edunderaunimodular
transformationofMleavingthesetsMuand{ed}invariant.
WithoutrestrictionweidentifynowPandL(P),inparticularu=−md∗.
Letv∈V(P)andwritev=jd=1qjaj=jd=1qjmjasrationallinear
combinations.Wesett:=u,v∈{−1,...,w−1}.
Forj=1,...,d−1wehaveqj=ηFjαj,v−t∈[−w,w].
Wegetthetwoinequalitiesforcintwodifferentways:
1.ToboundthecoefficientsofverticesofPintheZ-basism1,...,mdofM
wecanassumethatthecoefficientqjiseitherminimalormaximalamong
q1,...,qd−1.
(a)Aj,j=w.Assumej>1andthereisk<jmaximalwithAj,k>0.
Sincewmkiscontainedinthecolumnspace,thereexistsn∈Nsuch
thatAkw,kAj,k=nw,hence0≤Aj,k=nAk,k<Ak,k,son=0and
Aj,k=0,acontradiction.Thisyieldsqj=wqj∈[−w2,w2],hence
|qj|≤w((d−2)(w−1)+z),sinced≥3.
(b)Aj,j<w,i.e.,Aj,j≤z.Wehaveqj=kj=1Aj,kqk.Whenqjis
Whenmaximal,qwisegetminimal,qj≤(wde−get2)(qw−≥1)(wd+−z2)(ww=−w((1)(w−−w)1)(+dz−(−2)w+)z=).
(−w)((wj−1)(d−2)+z).Thisjgivesthefirstbound.
2.Firstwesubtractfori=1,...,d−1fromtheithrowofAA2i,i-times
{−thedwth,.ro..w,ofwA}..ByThisthisgivesproacedurematrixwhaevingfindaasZen-basistriesofonlyMinsuchtegersthatin
22a1,...,ad∈2wE1d.


160Chapter6.Centrallysymmetricreflexivepolytopes

Secondwecanassumethatqd=−t−jd−=11qjischosentobemaximal
αjorminimal.Thenqd=−t−jd−=11ηFj,v−t.Ifqdismaximal,thisyields
qd≤(d−2)(w−1)+(d−1)=(d−2)w+1.Ifqdisminimal,weget
qd≥(d−2)(−1)−(d−1)(w−1)=−((d−1)w−1).
Inanycaseweobtainq1,...,qd∈[−((d−1)w−1),(d−1)w−1].This
givesthesecondbound.

cenFtrallyorw=2symmetricthesecondreflexivbepoundolytopisalwesahaysvingbettersuchthanavtheertexfirst,canitbeemyieldsbeddedthat
intothe(2d−3)thmultipleoftheunitlatticecube.Wegotsignificantimprove-
mentsofthisresultinthespecialcasesofPropositions6.4.1,6.4.2,6.4.3.Also
notesecondthat(c≤for6dw−=9).3thefirstboundinthetheoremisalwaysbetterthanthe
wasUsuallydefinedoneindotheesnotfollodealwingwithproptheosition:invariantwbutwiththesocalledwidth
Proposition6.4.5([Deb03]).LetPbeacanonicalFanopolytope.
setWew:=max{u,v+1:u∈V(P∗),v∈V(P)}.
Thenvol(P)≤(w)d.
canonicalEspeciallyFanowpehaolytopveaeresult,couldbethatemwbouldeddedbeintotrivial,theifunitanycenlatticetrallycube.symmetric
Corollaryd6.4.6.AnycentrallysymmetriccanonicalFanopolytopehasvolume
.2mostatyields:3.7.14TheoremHencededinCorollarythelattic6.4.7.ecubAenyofdside-dimensionallengthatcmostanonicddal!(Fw)anod,pwherolytopewePiscandefinebedembaseind-
6.4.5.NowthisshouldbecomparedwiththefollowingcorollarytoTheorem6.4.4:
Corollary6.4.8.Anyd-dimensionalreflexivepolytopewithafacetFhavinga
cubvertexeofthatsideislengthonlycatontainemostdd(inw(!)d2−,1)wherfaceetswofisFcdefineandbeasembinedde6.4.5.dinthelattice
Weseethatespeciallyforsmallvaluesofwthisboundisasharpeningof
one.previousthe

tsoinpLattice6.5Byanembeddingwetriviallygetthatthenumberoflatticepointsinthepoly-
topeisboundedby3dwithequalityonlyinthecaseoftheunitlatticecube.
Howeverthisiseventrueingeneral:

tsoinpLattice6.5.

161

Theorem6.5.1.LetP⊆MRbeacentrallysymmetriccanonicalFanopolytope.
Then|P∩M|≤3d.
1−dAnyIffacPetisofPadditionalhasatlyrmosteflexive,3thenlatticethepfoloints.lowingstatementsareequivalent:
1.|P∩M|=3d
2.EveryfacetofPhas3d−1latticepoints
3.P=∼E1daslatticepolytopes
Weevenhavethefollowingalgebraic-geometricformulation:
Proposition6.5.2.LetX=X(N,)beacompleteGorensteintoricvariety
withcentrallysymmetric(1).Then
h0(X,−KX)≤3d.

Theproofsoftheseboundsrelyonthemethodofcountingmodulo3.Sowe
mapthedefineα:P∩M→M/3M=∼(Z/3Z)d.
ProofofProposition6.5.2.Leth∈SF(N,)definetheCartierdivisor−KX,
henceP:=Ph:={x∈MR:vτ,x≥−1∀τ∈(1)}.Since(1)iscen-
trallysymmetric,wegetthatPisacentrallysymmetricrationalpolytopeand
|P∩M|=h0(X,−KX)by(1.7).Henceweonlyhavetoshowthatαisinjec-
tive.Sosupposetherearex,y∈P∩Msuchthatα(x)=α(y).Thisimplies
(x−y)/3∈M.Forarbitraryτ∈(1),weget
vτ,x−vτ,y=vτ,(x−y)/3∈Z.
3Sincebyassumptionvτ,x,vτ,y∈{−1,0,1},thisyieldsvτ,x=vτ,yfor
anyτ∈(1).Thereforex=y,becauseiscomplete.
Proofoftheorem6.5.1.Lettherebex,y∈P∩Msuchthatα(x)=α(y),hence
(x−y)/3∈intP∩M={0},sox=y.Thereforeαisinjective.∗
LetF∈F(P)bearbitrarybutfixed.Defineu:=ηF∈V(P)andalso
theZ/3Z-extendedmapα(u):M/3M→Z/3Z.Form∈P∩Mwehave
u,m∈{−1,0,1},inparticular
m∈F⇐⇒α(u),α(m)=−1∈Z/3Z.
rivial.T1.:3.⇒1.⇒2.:IfPcontains3dlatticepoints,thenαisabijection,andtherefore
|F∩M|=|{z∈M/3M:α(ηF),z=−1}|=3d−1.

162Chapter6.Centrallysymmetricreflexivepolytopes

2.⇒3.:TheassumptionimpliesthatforanyfacetF∈F(P)themap
α|F:F∩M→{z∈M/3M:α(ηF),z=−1}
isabijection.Definex:=(1/3d−1)m∈F∩Mm∈relintF.
ByTheorem6.1.1(3)itremainstoprovethatxisaroot,i.e.,x∈M.
ChooseafacetG∈F(P∗)andanR-linearlyindependentfamilyw1,...,wd
ofverticesofGsuch∗thatw1=uandw2,...,wdarecontainedina(d−2)-
.PoffacedimensionalDenotethecorrespondingfacetsofPbyF1,F2,...,FdwithηFj=wjfor
j=1,...,d,soF1=F.ThenQ:=∩jd=2Fjisaone-dimensionalfaceofP.
Thereforealsotheaffinespanofα(Q∩M)isaone-dimensionalaffinesubspace
ofM/3M.Since|F∩Q|=1thereexistsanelementb∈M/3Msuchthat
α(u),b=0andα(wj),b=−1forallj=2,...,d.Applyingtheassumption
toF2yieldsalatticepointv∈P∩Mwithα(v)=b.Hencealsou,v=0and
wj,v=−1forj=2,...,d.
By3.1.4(11)wefindaZ-basise1∗=u,e2∗,...,ed∗ofNsuchthatforany
∙Fact1:wk,m∈F∩Mm=0fork=2,...,d.
j=2,...,dthereexistλj,k∈Rwithej∗=λj,2(w2−u)+∙∙∙+λj,d(wd−u).
(Proof:SinceF∩Fk=∅,theassumptionimpliesfori=−1,0,1∈Z/3Z:
∙Fact2:k=2λj,k∈Zforj=2,...,d.
|{z∈M/3Md:α(u),z=−1,α(wk),z=i}|=3d−2.)
(Proof:ej∗,v=(−kd=2λj,k)u,v+kd=2λj,kwk,v=−kd=2λj,kbythe
.)vofhoicecUsingthesetwofactswecanfinishtheproof:
e1∗,x=u,x=−1∈Z,
ej∗,x=(1/3d−1)(−λj,k)u,m+λj,kwk,m
dd
k=2m∈F∩Mk=2m∈F∩M
d=λj,k∈Zforj=2,...,d.
=2k.MxHence∈InthesimplicialcasethelasttheoremofthisthesisshowsthatDdhasthe
maximalnumberoflatticepointsamongallsimplicialreflexivepolytopesthat
haveacentrallysymmetricpairoffacets:
Theorem6.5.3.LetP⊆MRbeasimplicialreflexivepolytopewithacentrally
symmetricpairoffacets.Then
|P∩M|≤2d2+1.
AnyfacetofPhasatmost(d2+1)dlatticepoints.
Thefollowingstatementsareequivalent:
1.|P∩M|=2d2+1
2.Somefacethas(d2+1)dlatticepoints

6.5.tsoinpLattice

163

3.Anyfacethas(d2+1)dlatticepoints
4.P=∼Ddaslatticepolytopes
Proof.LetF,−F∈F(P).ApplyingTheorem6.3.1toPandFwecanassume
thatV(F)={e1,...,ed}arethecolumnsofamatrixAoftheformgivenin
thetheorem.Henceweget:
m∈(F∩M)\V(P)=⇒m=ei+ejfor(ei)i=2.(6.1)
2particularIn|F∩M|≤d+d=(d+1)d,
2202id1−dwhereequalityimplies
A=1∙∙∙11,
henceRemark6.3.11impliesP=∼Dd.Inparticularthisproves2.⇒3.⇒4.,
sincebyCor.6.3.4allfacetsofPhavethesamenumberoflatticepoints.
Letu:=ηFandm∈∂P∩M∩u⊥.By3.5.6wegetm∈Fiforsome
i∈{1,...,d}.Ifm∈V(P),thenwehavem∈{v1,...,vd}.Soletm∈V(P).
Inparticularvi∈u⊥,sovi=−ei.Sincemisnotawayfromvi,weeasily
obtainonlybylookingatQ:=lin(ei,m)∩PthatQ=∼E12aslatticepolytopes
(inthee+zlattice<ei,m>Z=lin(ei,m)∩M).Letz:=z(m,ei)∈V(Q)∩F∩M.
Sincei2∈F∩M,equation(6.1)impliesthatz=ejforsomej=i.Hence
m=−ei+ej.
2yieldsthisparticularIn|∂P∩M∩u⊥|≤d(d−1).
whereequalityholdsforP=∼Dd.
Hencewehave
|P∩M|≤1+2|F∩M|+d(d−1)=2d2+1,
whereequalityimplies|F∩M|tobemaximal.Thisprovestheimplications
4.⇒1.⇒2..

Index

31-divisor,QV(P)-primitivecollection,91
(Q-)factorialvariety,globally-,31
5424-cell,AnAdditiveticanonicalvertex,divisor,9832
98t,oinAS-p122e,reductivgroup,Automorphism47from,yaAw25ter,Barycen20orus,TBigBlaschke-Santal´oinequality,25
divisorCanonical32general,35toric,divisorCartier28ample,27t-free,oinbase-p26big,26nef,22t,arianvtorus-in28ample,eryv21group,ClassClassnumber,21
69lattice,Coarsest19cone,dualCone,97tiguous,ConConvexhull,19
54e,olytopCrosspCurves,torus-invariant,23
Degree40ticanonical,an28edding,bemofofofvertex,primitiv97erelation,92
ezzoPDelpolytopes,varieties,153
57toric,surfaces,

165

31Desingularization,Determinantofaffinesublattice,25
34,Discrepancy88diameter,Discrete97t,oinpDoubleEdge,Ehrhart,23polynomial,theorem,reci-
procitylaw,25
119metric,ahlerEinstein-K¨157edding,bEmesterSylvEnlarged75partition,73system,teighwExceptional33divisors,32cus,lo19ace,F23acet,F123basis,acetFanF21complete,19general,of24normals,122semisimple,35simplicial,spannedbypolytope,24
Fanopcanonical,olytopeterminal,smooth,38
94oth,quasi-smoFanovgeneral,ariety36
39Gorenstein,40toric,Gorenstein,37toric,36eak,w37eak,toric,w26sections,Global33index,Gorenstein

166

GorensteintoricFanovariety,40
23theorem,Helly’sHermiteHomogeneousnormalcoform,ordinate69ring,123
23e,relativterior,InIntersectionnumber,27
86unit,e,cubLatticeLatticeLatticelengthdistance,of44edge,86
Latticepointenumerator,25
19lattice,dualLattice,46Link,23cone,Mori21t,arianequivMorphism,39MPCP-Desingularization,32cus,loNonsingularNormal19general,uniqueuniqueinner,primitiv24einner,25
24cone,Normal24fan,Normal22group,PicardPicardnumber,22
PPick’solyhedron,theorem,2326
eolytopPdual,empty,2445
23general,23rational,lattice,24ducts,proreflexivreflexive,e,43semisimple,130
49semi-terminal,24simple,24simplicial,138symmetric,Positivehull,19
collectionePrimitiv90fan,ofofreflexivepolytope,91
Primitivelatticepoint,22
92relation,ePrimitiv

Index

PseudodelPezzopolytopes,vari-
153eties,eQuasi-primitiv95collection,95relation,Quasi-smoothFanopolytope,94
ulaformRamification33general,36toric,21ys,RaRefinement,crepant,coherent,39
59dimension,eReflexivReflexivepolytope,43
Reflexiveweightsystem,71
singularitiesofResolutioncrepancoherent,t33crepant,39
crepangeneral,t,31toric,36
RoRoototsbasis,123
122general,unipsemisimple,otent,122122
equivorthogonal,alent,124123
32crossings,normalSimpleSimpleSimplex,p24oint,97
Singularquasi-smoothFanopolytope,
94Singularities34terminal,canonical,89conifold,logresolutioncanonical,of,31logterminal,34
49semi-terminal,36toric,othSmoFanopolytope,38
94parallelogram,lattice94triangle,StarSpanningset,46latticesimplex,69
StellarSteinitz’ssubtheorem,divisions,2336
Strictlyupperconvex,28

Index

SuppSupportortoffunction,fan,2123
esterSylv75partition,73sequence,Symmetricweightvertex,system,9873

21w-up,blooricTToricFanovariety,37
21fibre-bundle,oricTToricQvariet-factorial,y35
20affine,35nonsingular,factorial,20general,39Gorenstein,20fan,ofofpropjectivolytope,e,2829
138symmetric,

partition,Unit74Upperconvex,27

77t,constanardiV23ertex,V25e,relativolume,V

23alls,WsystemteighWassociatedtopolytope,69
68general,68normalized,68reduced,68of,reductionWeightedbarycenter,140
Weightedprojectivespace,69
Weildivisor,torus-invariant,21
160Width,

e,Zonotop58lattice,standard

167

yBibliograph

[AKMS97]Avram,A.C.;Kreuzer,M.;Mandelberg,M.;Skarke,H.:Searching
forK3Fibrations.Nucl.Phys.B494,567-589(1997)
[AS70]Aho,A.V.;Sloane,N.J.A.:Somedoublyexponentialsequences.
FibonacciQuart.11,429-437(1970)
[Bat82a]Batyrev,V.V.:ToroidalFano3-folds.Math.USSR-Izv.19,13-25
(1982)[Bat82b]Batyrev,V.V.:Boundnessofthedegreeofhigher-dimensionaltoric
Fanovarieties.MoscowUniv.Math.Bull.37,28-33(1982)
[Bat84]Batyrev,V.V.:Higher-dimensionaltoricvarietieswithampleanti-
canonicalclass.PhDThesis(inRussian).MoscowStateUniversity
1984[Bat91]Batyrev,V.V.:Ontheclassificationofsmoothprojectivetoric
varieties.TohokuMath.J.43,569-585(1991)
[Bat94]Batyrev,V.V.:DualpolyhedraandmirrorsymmetryforCalabi-
Yauhypersurfacesintoricvarieties.J.Algebr.Geom.3,493-535
(1994)[Bat99]Batyrev,V.V.:OntheclassificationoftoricFano4-folds.J.Math.
Sci.NewYork94,1021-1050(1999)
[BB93]Borisov,A.;Borisov,L.:SingulartoricFanovarieties.Math.USSR
(1993)277-283,75[BC94]Batyrev,V.V.;Cox,D.A.:OntheHodgestructureofprojective
hypersurfacesintoricvarieties.DukeMath.J.75,293-338(1994)
[BF71]Bonnesen,T.;Fenchel,W.:TheoriederkonvexenK¨orper.Provi-
dence,RI:AMSChelseaPublishingCompany1971
[BG99]Bruns,W.;Gubeladze,J.:Polytopallineargroups.J.Alg.218,
(1999)715-737[BG02]Bruns,W.;Gubeladze,J.:PolyhedralK2.ManuscriptaMath.109,
(2002)367-404[BK85]Betke,U.;Kneser,M.:ZerlegungenundBewertungenvonGitter-
polytopen.J.ReineAngew.Math.358,202-208(1985)

169

170

[Bor00][BS99]uh96]¨[B[BV97][Cas03a][Cas03b][Cas03c][Cas04][CK99][Con02]x95][Cox97][Cox03][Co[Cur22][Dai02][Deb01]

yBibliograph

Borisov,A.:Convexlatticepolytopesandconeswithfewlat-
ticepointsinside,fromabirationalgeometryviewpoint.Preprint,
(2000)G/0001109math.ABatyrev,V.V.;Selivanova,E.N.:Einstein-K¨ahlermetricsonsym-
metrictoricFanomanifolds.J.ReineAngew.Math.512,225-236
(1999)B¨uhler,D.:HomogenerKoordinatenringundAutomorphismen-
gruppevollst¨andigertorischerVariet¨aten.Diplomarbeit(inGer-
man).Math.Inst.derUniversit¨atBasel1996
Brion,M.;Vergne,M.:LatticePointsinSimplePolytopes.J.Am.
Math.Soc.10,371-392(1997)
Casagrande,C.:ToricFanovarietiesandbirationalmorphisms.
Int.Math.Res.Not.27,1473-1505(2003)
Casagrande,C.:CentrallysymmetricgeneratorsintoricFanova-
rieties.Manuscr.Math.111,No.4,471-485(2003)
Casagrande,C.:Contractibleclassesintoricvarieties.Math.Z.
(2003)99-126,243Casagrande,C.:ThenumberofverticesofaFanopolytope.
(2004)G/0411073math.At,PreprinCox,D.A.;Katz,S.:Mirrorsymmetryandalgebraicgeometry.
Mathematicalsurveysandmonographs68.Providence,RI:Amer.
1999c.SoMath.Conrads,H.:Weightedprojectivespacesandreflexivepolytopes.
ManuscriptaMath.107,215-227(2002)
Cox,D.A.:Thehomogeneouscoordinateringofatoricvariety.J.
Algebr.Geom.4,17-50(1995)
Cox,D.A.:Recentdevelopmentsintoricgeometry.In:Algebraic
geometry(SantaCruz,1995),389-436,Proc.Sympos.PureMath.
62,Part2.Providence,RI:Amer.Math.Soc.1997
Cox,D.A.:Whatisatoricvariety?.Manuscriptonwebpage,
http://www.amherst.edu/∼dacox/(2003)
Curtiss,D.R.:OnKellogg’sdiophantineproblem.Amer.Math.
Monthly29,380-387(1922)
Dais,D.I.:Resolving3-dimensionaltoricsingularities.In:Geom-
etryoftoricvarieties(Grenoble,2000),155-186,S´emin.Congr.6.
Paris:Soci´ete´Math´ematiquedeFrance2002
Debarre,O.:Higher-dimensionalalgebraicgeometry.Universitext.
NewYork:Springer-Verlag2001

Bibliography

[Deb03][DHZ01][Eik92][Eik93][Eis94][Epp04]e88][Ev[EW91]a88][Ewa96][Ewuj03][Ful93][F[GJ00][GJ05][GKP89]

171

Debarre,O.:Fanovarieties.In:Higherdimensionalvarietiesand
rationalpoints(Budapest,2001),93-132,BolyaiSocietyMathe-
maticalStudies12.Berlin:Springer-Verlag2003
Dais,D.I.;Haase,C.;Ziegler,G.M.:Alltoricl.c.i.-singularities
admitprojectivecrepantresolutions.Preprint,math.AG/9812025
(1998)Eikelberg,M.:ThePicardgroupofacompacttoricvariety.Results
inMath.22,509-527(1992)
Eikelberg,M.:Picardgroupsofcompacttoricvarietiesandcom-
binatorialclassesoffans.ResultsinMath.23,251-293(1993)
Eisenbud,D.:Commutativealgebrawithaviewtowardalge-
braicgeometry.GraduateTextsinMathematics150.NewYork:
1994erlagSpringer-VEppstein,D.:Egyptianfractions.Webpage,
http://www.ics.uci.edu/∼eppstein/numth/egypt/(2004)
Evertz,S.:ZurKlassifikation4-dimensionalerFano-Variet¨aten.
Diplomarbeit(inGerman).Math.Inst.derRuhr-Universit¨at
1988umhcBoEwald,G.;Wessels,U.:Ontheamplenessofinvertiblesheavesin
completeprojectivetoricvarieties.ResultsinMath.19,275-278
(1991)Ewald,G.:OntheclassificationoftoricFanovarieties.Discrete
Comput.Geom.3,49-54(1988)
Ewald,G.:Combinatorialconvexityandalgebraicgeometry.Grad-
uateTextsinMathematics168.NewYork:Springer-Verlag1996
Fujino,O.:NotesontoricvarietiesfromMoritheoreticviewpoint.
TohokuMath.J.55,551-564(2003)
Fulton,W.:Introductiontotoricvarieties.AnnalsofMathematics
Studies131.Princeton,NJ:PrincetonUniversityPress1993
Gawrilow,E.;Joswig,M.:polymake:Aframeworkforanalyzing
convexpolytopes.In:Kalai,G.;Ziegler,G.M.:Polytopes-Com-
binatoricsandComputation.DMVSeminar29.Basel:Birkh¨auser
2000Gawrilow,E.;Joswig,M.:polymake:atoolforthealgorith-
mictreatmentofconvexpolyhedraandfinitesimplicialcomplexes.
Webpage,http://www.math.tu-berlin.de/polymake/(2005)
Graham,R.L.;Knuth,D.E.;Patashnik,O.:Concretemathemat-
ics:Afoundationforcomputerscience.Reading,MA:Addison-
1989esleyW

172

[GW93][Har77][Has00][Hen83][Hib92][HM04][IK95][Kan98][Kas03][Kas04][KKMS73][Kre03a][Kre03b][KS96][KS97][KS98][KS00][KS02]

yBibliograph

Gruber,P.;Wills,J.:HandbookofConvexGeometry,Vol.Aand
B.Amsterdam:ElsevierSciencePublishersB.V.1993
Hartshorne,R.:Algebraicgeometry.GraduateTextsinMathemat-
ics52.NewYork:Springer-Verlag1977
Haase,C.:Latticepolytopesandtriangulationswithapplications
totoricgeometry.PhDThesis.Math.Inst.derTechnischenUni-
2000Berlinatersit¨vHensley,D.:Latticevertexpolytopeswithinteriorlatticepoints.
PacificJ.Math.105,183-191(1983)
Hibi,T.:Dualpolytopesofrationalconvexpolytopes.combinator-
(1992)237-240,12icaHaase,C.;Melnikov,I.V.:Thereflexivedimensionofalattice
polytope.Preprint,math.CO/0406485(2004)
Izboldin,O.;Kurliandchik,L.:Unitfractions.AMSTransl.,Series
(1995)193-200,1662,Kantor,J.-M.:TriangulationsofintegralpolytopesandEhrhart
polynomials.Contrib.toAlg.andGeom.39,205-218(1998)
Kasprzyk,A.M.:ToricFano3-foldswithterminalsingularities.
(2003)G/0311284math.At,PreprinKasprzyk,A.M.:TheFanopolytopesinZ3.pdf-file,
http://www.maths.bath.ac.uk/∼mapamk/pdf/Fanolist.pdf
(2004)Kempf,G.;Knudsen,F.;Mumford,D.;Saint-Donat,B.:Toroidal
embeddings.LectureNotesinMath.339.NewYork:Springer-
1973erlagVKreuzer,M.:Einstein-KaehlerMetrics.Personalmanuscript(2003)
Kreuzer,M.:SomeBasicsabouttheEhrhartPolynomial;The
BarycentricGeneralization.Personalmanuscript(2003)
Kreuzer,M.;Skarke,H.:Reflexivepolyhedra.Script,Trieste
SpringSchoolandWorkshop(1996)
Kreuzer,M.;Skarke,H.:Ontheclassificationofreflexivepolyhe-
dra.Commun.Math.Phys.185,495-508(1997)
Kreuzer,M.;Skarke,H.:Classificationofreflexivepolyhedrain
threedimensions.Adv.Theor.Math.Phys.2,853-871(1998)
Kreuzer,M.;Skarke,H.:Completeclassificationofreflexivepoly-
hedrainfourdimensions.Adv.Theor.Math.Phys.4,1209-1230
(2000)Kreuzer,M.;Skarke,H.:Reflexivepolyhedra,weightsandtoric
Calabi-Yaufibrations.Rev.Math.Phys.14,343-374(2002)

yBibliograph

[KS04a][KS04b][Lat96][Lut93][LZ91][Mab87][Mat02]ul01]¨[M[Nam97][Nil04a][Nil04b][Nil04c][Oda78][Oda88]y04]a[P[Pik00]

173

Kreuzer,M.;Skarke,H.:PALP:Apackageforanalyzinglattice
polytopeswithapplicationstotoricgeometry.ComputerPhys.
(2004)87-106,157Comm.,Kreuzer,M.;Skarke,H.:Calabi-Yaudata.Webpage,
http://hep.itp.tuwien.ac.at/∼kreuzer/CY(2004)
Laterveer,R.:Linearsystemsontoricvarieties.TohokuMath.J.
(1996)451-458,48Lutwak,E.:Selectedaffineisoperimetricinequalities.In:Gruber,
P.M.;Wills,J.M.:Handbookofconvexgeometry,VolumeA.Am-
1993PublishersScienceElseviersterdam:Lagarias,J.C.;Ziegler,G.M.:Boundsforlatticepolytopescon-
tainingafixednumberofinteriorpointsinasublattice.Can.J.
(1991)1022-1035,43Math.Mabuchi,T.:Einstein-K¨ahlerforms,Futakiinvariantsandconvex
geometryontoricFanovarieties.OsakaJ.Math24,705-737(1987)
Matsuki,K.:IntroductiontotheMoriprogram.Universitext,
(2002)orkNew-Yerlag,Springer-VM¨uller,A.:TorischeFano-Variet¨atenmitconifoldSingularit¨aten.
Diplomarbeit(inGerman).Math.Inst.derEberhard-Karls-
Universit¨atT¨ubingen2001
Namikawa,Y.:SmoothingFano3-folds.J.Alg.Geom.6,307-324
(1997)Nill,B.:GorensteintoricFanovarieties.Preprint,
math.AG/0405448.Toappearin:Manuscr.Math.116,183-
(2005)210Nill,B.:Completetoricvarietieswithreductiveautomorphisms
group.Preprint,math.AG/0407491(2004)
Nill,B.:Volumeandlatticepointsofreflexivesimplices.Preprint,
(2004)G/0412480math.AOda,T.:Lecturesontorusembeddingsandapplications.(Basedon
jointworkwithKatsuyaMiyake.).TataInstituteofFundamental
ResearchLecturesonMathematicsandPhysics,Mathematics57.
NewYork:Springer-Verlag1978
Oda,T.:Convexbodiesandalgebraicgeometry-Anintroduction
tothetheoryoftoricvarieties.ErgebnissederMathematikund
ihrerGrenzgebiete15.Berlin:Springer-Verlag1988
Payne,S.:Fujita’sveryamplenessconjectureforsingulartoric
varieties.Preprint,math.AG/0402146(2004)
Pikhurko,O.:Latticepointsinsidelatticepolytopes.Preprint,
(2000)math.CO/0008028

174

[PR00][Pro04][PSG99][PWZ82][Rei83][Sat00][Sat02][Slo04][Sta86][Sti01]ah71][T[VK85]ag95][W[Wir97][WW82][WZ04][Zie95][Zie01]

yBibliograph

Poonen,B.;Rodriguez-Villegas,F.:Latticepolygonsandthenum-
ber12.Am.Math.Soc.Monthly107,238-250(2000)
Prokhorov,Y.G.:OnthedegreeofFanothreefoldswithcanonical
Gorensteinsingularities.Preprint,math.AG/0405347(2004)
Parshin,A.N.;Shafarevich,I.R.;Gamkrelidze,R.V.:AlgebraicGe-
ometryV:FanoVarieties.EncyclopaediaofMathematicalSciences
1999erlagSpringer-VBerlin:47.Perles,M.;Wills,J.;Zaks,J.:Onlatticepolytopeshavinginterior
latticepoints.ElementederMathematik37,44-46(1982)
Reid,M.:Decompositionoftoricmorphisms.In:Arithmeticand
geometry,Vol.II:Geometry.ProgressinMathematics36.Boston:
1983auserBirkh¨Sato,H.:Towardtheclassificationofhigher-dimensionalFanova-
rieties.TohokuMath.J.,52,383-413(2000)
Sato,H.:StudiesontoricFanovarieties.TohokuMath.Publ.23,
TohokuUniversity(2002)
Sloane,N.J.A.:On-lineencyclopediaofintegersequences.Web-
page,http://www.research.att.com/∼njas/sequences/(2004)
Stanley,R.P.:EnumerativeCombinatorics,Vol.I.Monterey,CA:
Wadsworth&Brooks/Cole1986
Stillwell,J.:Thestoryofthe120-cell.NoticesoftheAm.Math.
Soc.48,17-24(2001)
Tahara,K.:OnthefinitesubgroupsofGL(3,Z).NagoyaMath.J.
(1971)169-209,41Voskresenskij,V.E.;KlyachkoA.A.:ToroidalFanovarietiesand
rootsystems.Math.USSR-Izv.24,221-244(1985)
Wagner,H.:GewichteteprojektiveR¨aumeundreflexivePoly-
tope.Diplomarbeit(inGerman).Math.Inst.derRuhr-Universit¨at
1995umhcBoWirth,P.R.:ZentralsymmetrischereflexivePolytope.Diplomar-
beit(inGerman).Math.Inst.derRuhr-Universit¨atBochum1997
Watanabe,K.;Watanabe,M.:TheclassificationofFano3-folds
withtorusembeddings.TokyoJ.Math.5,37-48(1982)
Wang,X.-J.andZhu,X.:K¨ahler-Riccisolitonsontoricmanifolds
withpositivefirstChernclass.Adv.Math.188,87-103(2004)
Ziegler,G.M.:Lecturesonpolytopes.Graduatetextsinmathe-
matics152.NewYork:Springer-Verlag1995
Ziegler,G.M.:Questionsaboutpolytopes.In:Engquist,B.;
Schmid,W.:Mathematicsunlimited-2001andbeyond.Berlin:
2001erlagSpringer-V

-AendixAppinZusammenfassungSpracherdeutsche

deltDieseesArbsicheitumbvefasstollst¨sichandigemitnormaletorischentorischeGorenstein-FVariet¨aten,ano-Varietderen¨aten.antikDabanonisceihan-her
ist.CartierdivisoramplereinDivisorDiesealgebraisch-geometrischenObjektehabenihreEntsprechunginder
keinonvexenreflexivesPGeometrieolytopineinderFGitterpormsoolytop,genanndasterinseinemreflexiverPInnerenolytopdene.DabUrsprungeiist
enlytopth¨alt,ist.mitInsbderesondereEigenschaft,tretendassreflexivdasePdualeolytopPeolytopimmerinwiederumPaareneinauf.GitterpDie-o-
seBegriffsbildungwurdeerstmalsvonBatyrevin[Bat94]eingef¨uhrt,alser
Vzeigte,ariet¨atendassCalabi-YgeneriscauheansindtikundanoniscsichhenachHypAuferfll¨¨acosunghendertorischerSingularit¨atenGorenstein-Faufgrundano-
dernat¨urlichenDualit¨atreflexiverPolytopeKandidatenf¨urMirror-Symmetrie
ergeben.Daraufhinwurdeangestrebt,s¨amtlichereflexivePolytopeimphysi-
kfandenalischscrelevhließlicantenhmitHilfevierdimensionalenihresFallzuComputerprogrammsklassifizieren.PALPKreuzer[KS04aund]16,Skark4319e
bzw.lytope[KS98473800776,KS00nic,hKS04btisomorphe].Weiterzwei-,istbdrei-ekannbzw.t,dassesinvierdimensionalejederDimensionreflexivenPuro-
Yendlicau-Vharietviele¨atenhabenIsomorphieklassenKreuzerundgibt.SkAufarkederinSucletzterhenacZeithauchallgemeinerenbegonnen,fCalabi-¨unf
undsechsDimensioneninAngriffzunehmen[KS04b].
W¨ahrendesschoneinigemathematischeArbeitengibt,diesichmitglatten
torischenFano-Variet¨atenbesch¨aftigen[WW82,Bat82a,Bat82b,Bat99,Sat00,
tersucDeb03h,t,insbCas03a,esondereCas03bin],h¨wurdeoherendersingulDimensionen.¨areFallDiesnohc¨hangtnichtdamitsointensivzusammen,un-
chedasshieralgebraisceinigeh-geometriscfundamenhetaleMethoSchdenwierigkwieeitenRiemann-Roauftreten.choZunder¨achstScsindhnitttheorieman-
nenichttoriscohneheWkrepaneiteresteanAufwl¨osungendbar,insbexistierenesonderemuss.daZuminh¨ZwohereneitenverwDimensionenendenkvieleei-
konvex-geometrischeBeweisedieVoraussetzung,dassdieEckeneinerFacette
imglattenFalleineGitterbasisbilden,wogegenreflexivePolytopeimAllgemei-
vnenerhindertsogaralleinGitterpunktedieextremimgroßeInnerenAnzahleinerFanacetteenIsomorphieklassenthaltenk¨onnen.selbstScinhließlicniedri-h
diegenohnedieDimensionenHilfeineinesdenmeistenComputersF¨allenauskeineommvt.ollst¨andigerigoroseKlassifikation,

176

ZusammenfassungindeutscherSprache

DasZieldieserDissertationisteineerstesystematischeUntersuchungtori-
scherGorenstein-Fano-Variet¨aten.Hierf¨urwerdenzun¨achstMethodenundRe-
sultate¨uberglattetorischeFano-Variet¨atenauftorischeFano-Variet¨atenmit
mildenSingularit¨atenverallgemeinert.Dabeil¨asstdieKonzentrationaufMe-
thodenderkonvexenGeometrie¨auchschonimglattenFallbekannteResultate
transparentererscheinen.UberraschendistindiesemZusammenhangdieGut-
artigkeitQ-faktoriellertorischerGorenstein-Fano-Variet¨aten.Weiteristesdas
Ziel,inwichtigenF¨allenvollst¨andigeKlassifikationsresultateauchinh¨oherenDi-
metriscmensionenhenzuEigenscgewinnen.haftenDabundeiInvstehenariantendurchwreflexivegdieerkPomolytopebinatoriscimVhenundordergrund,geo-
f¨urwelcheEinschr¨ankungen,Absch¨atzungenundVermutungenbewiesenund
formuliertwerden.DieseErgebnisselieferninsbesondereauchErkl¨arungenf¨ur
interessanteBeobachtungeninderDatenbank.
DieseArbeitbestehtauseinerEinleitung,einerListederverwendetenNota-
tionen,sechsKapitelnsowieeinemIndexundeinerausf¨uhrlichenBibliographie.
Jedesgr¨oßereKapitelbesitzteineeigeneEinleitung,dereine¨Ubersichtslistemit
ReferenzenaufdiewichtigstenErgebnisseangef¨ugtist.
ImFolgendensollennundieeinzelnenKapiteldieserArbeitzusammenfas-
sendbeschriebenwerden.
IndenerstenbeidenKapitelnwerdenGrundlagenbesprochen.Kapitel1
enth¨altdiefundamentalenAussagendertorischenGeometrie.Kapitel2besch¨af-
tigtsichzun¨achstmitderAufl¨osungundHierarchievonSingularit¨atenundde-
renBeschreibungimtorischenFall.SchließlichwerdentorischeFano-Variet¨aten
genaueruntersucht.Diesealgebraisch-geometrischenObjektehabeneineEins-
zu-eins-EntsprechunginderkonvexenGeometrieinFormsogenannterFano-
Polytope.TorischeFano-Variet¨atenmitkanonischenbzw.terminalenSingula-
rit¨atenkorrespondierendabeimitkanonischenbzw.terminalenFano-Polytopen.
Glattebzw.Q-faktorielletorischeFano-Variet¨atenentsprechenglattenbzw.sim-
plizialentorischenFano-Variet¨aten.
Kapitel3istdasHerzst¨uckdieserArbeit.HierwerdenreflexivePolytopede-
finiertunduntersucht,alsodiekonvex-geometrischenGegenst¨uckezutorischen
Fano-Variet¨atenmitGorenstein-Singularit¨aten,kurz,torischenGorenstein-Fano-
aten.¨arietVDasKapitelbeginntmitderVerallgemeinerungzweierwichtigerHilfsmittel,
dieschonerfolgreichzurUntersuchungglattertorischerFano-Variet¨atenein-
gesetztwurden.ZumEinenistdiesdieProjektionreflexiverPolytopeentlang
vonEckenoder,allgemeiner,entlangvonGitterpunktenaufdemRand.Diese
AbbildunghateinigewichtigeEinschr¨ankungen,dieinProposition3.2.2ge-
naubeschriebenwerden.HiersollnureineunmittelbareAnwendungerw¨ahnt
werden:Batyrevbewiesin[Bat99],dassdieantikanonischeKlasseeinesTorus-
invariantenPrimdivisorseinerglattentorischenFano-Variet¨atnumerischeffek-
tivist.Inkonvex-geometrischerSprachebesagtdies,dassdieProjektioneines
glattenFano-Polytopsreflexivist.HierwirdnuneineinfachererBeweisdieser
Aussageangegeben(Korollar3.2.8)unddar¨uberhinausverallgemeinerndin
Proposition3.2.4gezeigt,dassdieProjektioneinesterminalenreflexivenPoly-
topsentlangeinerEckeeinkanonischesFano-Polytopist.
BeidemzweitenwichtigenHilfsmittelhandeltessichumprimitiveKollek-
tionenundRelationen.DiessindspezielleMengenundRelationenvonEcken,

ZusammenfassungindeutscherSprache

177

dievonBatyrevin[Bat91]eingef¨uhrtwurden,umglatteFano-Polytopeeindeu-
tigglatterzubesctorischreibheren.FSieano-Vwarietaren¨atenessenintiellf[Bat99¨ur].seineDieseKlassifikationBegriffsbildungbenvierdimensionaler¨otigtje-
dochdieExistenzvonGitterbasenaufdenFacetten,istalsoimAllgemeinen
nichtaufreflexivePolytopeanwendbar.Beschr¨anktmansichaberaufdenFall
einerprimitivenKollektionderL¨angezwei,waseinfacheinemPaarvonEcken
entspricht,dienichtineinergemeinsamenFacetteliegen,sokanninProposition
3.3.1gezeigtwerden,dasseinesinnvolleVerallgemeinerungdesBegriffeseiner
primitivenRelationexistiert.AlseinedirekteAnwendungergibtsichinKorollar
3.3.2,dassjezweiEckeneinessimplizialenreflexivenPolytopsdurchmaximal
dreiKantenverbundenwerdenk¨onnen,wobeinurimFalleineszentralsymmetri-
schenPaaresm¨oglicherweisenichtschonzweioderwenigerKantenausreichen.
DerKantengraphhatalsoh¨ochstensDurchmesserdrei.MitHilfedieserkom-
binatorischenBeschr¨ankungenkannindenKorollaren3.3.4und3.3.5erkl¨art
werden,welchePlatonischenK¨orper¨uberhauptalsreflexivePolytopeauftreten
klei¨onnen.HinsichtPrimitivfundamenetalKollektionenundwurdenunddaherRelationenimderglattenL¨FangeallvzwoneisindCasagrandeinvieler-in
glatten[Cas03a]toriscgenauerhenFunano-Vtersuchariett.¨atSiediekonntePicardzahldortzeigen,einesTdassorusindievarianPicardzahltenPrimdi-einer
visorsh¨ochstensumdrei¨uberschreitenkann.MitHilfeobigerVerallgemeine-
rungenkannnuninKorollar3.5.17bewiesenwerden,dassdieentsprechende
Aussageauchf¨urQ-faktorielletorischeGorenstein-Fano-Variet¨atengilt.
DerHauptteildesdrittenKapitelsbesch¨aftigtsichdamit,obereSchranken
f¨urInvariantenreflexiverPolytopezufinden,f¨urdasVolumen,dieAnzahlder
EckenundderGitterpunkte.
F¨urdieEckenanzahleinesreflexivenPolytopsstellenwir,motiviertdurch
BeobachtungeninderDatenbank,dieVermutungauf,dasseind-dimensionales
reflexivesPolytoph¨ochstens6d/2Eckenhat,wobeiderMaximalfalleindeu-
tigbestimmtist.DieseVermutungwirdimletztenKapiteldieserArbeitf¨ur
zenHierdagegentralsymmetriscliegthedereinfacSchwheerpunktreflexiveaufPderolytopebmaximaleneliebigerEckDimensionenanzahlbsimplizia-ewiesen.
lerreflexiverPolytope.F¨urglatted-dimensionaleFano-Polytopewurdeschon
seitl¨angererZeitvermutet,dassdieseh¨ochstens3dEckenbesitzen,wobeidie
GleichheitnuringeradenDimensionenm¨oglichseinsollte.In3.5.7wirdnun
dieVermutungaufgestellt,dassdiesauchf¨ursimplizialereflexivePolytopegilt
undderFallvon3dEckendar¨uberhinauseindeutigbestimmtist.Unterder
Voraussetzung,dassdasdualePolytopeinzentralsymmetrischesEckenpaarbe-
sitzt,wirddieseVermutunginSatz3.5.11erstmalsbewiesen.Basierendaufder
Ver¨offentlichungdieserErgebnissealsPreprintin[Nil04a]istdieseVermutung
dannvonCasagrandevollst¨andigin[Cas04]gezeigtworden.
DesWeiterenwirdindiesemKapitelangestrebt,eineguteAbsch¨atzungf¨ur
dieAnzahlderGitterpunkteeinesreflexivenPolytopszufinden.Diesw¨urde
durcheineobereSchrankef¨urdasVolumenerreicht.DainderDatenbankdie
reflexivenPolytopemitdemgr¨oßtenVolumenallesamtSimplexesind,wurdein
dieserArbeitversucht,einescharfeAbsch¨atzungf¨urdasVolumeneinesreflexi-
venSimplexzufinden.DiesistderInhaltvonSatz3.7.13.Dieentsprechende
algebraisch-geometrischeAussagebesagt,dassjedetorischeGorenstein-Fano-
Variet¨atmitKlassenzahleins,alsoz.B.jedergewichteteprojektiveRaummit
Gorenstein-Singularit¨aten,h¨ochstenseinenantikanonischenGradvon9inDi-

178ZusammenfassungindeutscherSprache
mensionzwei,72inDimensiondreiund2(yd−1−1)2abDimensionvierbesitzt,
wobeidieMaximalf¨allegenaubestimmtwerdenk¨onnen.Diehierauftretende
Zahlenfolgey0,y1,y2,...wirddabeidurchy0=2undyn=1+y0∙∙∙yn−1de-
finiert.Eskannnunvermutetwerden,dassdieseobereAbsch¨atzungauchf¨ur
s¨amtlicheFano-Variet¨atenmitkanonischenGorenstein-Singularit¨ateng¨ultigist.
ImdreidimensionalenFallisteinesolcheVermutung,bekanntunterdemNa-
inmenFdiesemano-IskKapitelovvskikh-Verwermendetenutung,MethovordenKurzemgr¨undenin[Pro04darauf,]bdassewiesenjedemworden.reflexivDieen
SimplexeineMengevonStammbr¨uchenderSummeeins(z.B.21+31+61=1)
zugeordnetwerdenkann,siehe[Bat94]und[Con02].Konvex-geometrischeFra-
gestellungenf¨uhrensomitzunicht-trivialenProblemeninderelementarenZah-
vlenerwtheorie,andtdiesind.engWmiteiterhinderwirdDarstellunginSatz3.7.19rationalerbZahlenewiesen,alsdass¨esagyptiscinhejederBr¨ucDi-he
mensioneineneindeutigenreflexivenSimplexmitdermaximalenAnzahlan
GitterpunktenaufeinerKantegibt.Dieswarzuvorin[HM04]vonHaaseund
MelnikovinderDatenbankbeobachtetworden.
InKapitel4klassifizierenwirdreidimensionaletorischeFano-Variet¨atenmit
terminalenGorenstein-Singularit¨aten.DieseVariet¨atenentsprechendreidimen-
sionalenreflexivenPolytopen,dieaufdemRandaußerdenEckenkeineweiteren
Gitterpunktebesitzen.ImHauptsatz4.3.2diesesKapitelswirdgezeigt,dasses
genau100Isomorphieklassensolchersogenannterquasi-glatterFano-Polytope
gibt.DieIdeedesBeweisesist,mitdenMethodendesvorigenKapitelszuzei-
gen,dassquasi-glatteFano-Polytopeschondurchsogenanntequasi-primitive
Relationenbestimmtsind.DieKlassifikationselbstwirdexplizitausgef¨uhrt.
InKapitel5untersuchenwireinespezielleMengevonGitterpunkten,ge-
nanntWurzelmenge,diezueinemvollst¨andigenF¨acherassoziiertwerdenkann.
ImFalleeinesF¨achers,dervondenNormaleneinesreflexivenPolytopsauf-
imgespannInnerentwird,derFhandeltacettenesdessichPbeiolytops.denWDieurzelnRelevanzgeradederumWdieurzelmengeGitterpunkter¨uhrt
daher,dasssieessentiellf¨urdieBestimmungderAutomorphismengruppeei-
nertorischenVariet¨atist.SobestimmtdieAnzahlderWurzelndieDimension
derAutomorphismengruppe.BesonderswichtigsinddiejenigenWurzeln,deren
NegativeauchwiederWurzelnsind;siewerdenalshalbeinfachbezeichnet.Die
Automorphismengruppeistgenaudannreduktiv,wennjedeWurzelhalbeinfach
ist.IndiesemKapitelf¨uhrenwirsogenannteFacetten-Basenbzw.Wurzel-Basen
ein,dieaufgeometrischbefriedigendeWeisedieMengederFacetten,dieWur-
zelnenthalten,bzw.dieMengederhalbeinfachenWurzelnparametrisieren.F¨ur
denBeweisf¨urderenExistenzistimFallreflexiverPolytopederimvorigen
KapitelNutzen.vWiesicherallgemeinerteimNacBegriffhhineineinerherausstellte,primitivenentspracRelationhdieserderLin¨angediesemzweispvone-
ziellenZusammenhang¨UberlegungenvonBrunsundGubeladzein[BG99].Im
allgemeinerenFallvollst¨andigerVariet¨atenwerdenResultatevonCox¨uberden
homogenenKoordinatenringin[Cox95]herangezogen.AlsersteAnwendung
k¨jenigenonnenvwirollstin¨andigenPropositiontorischen5.1.19VarietPro¨atenduktecprojektivharakterisieren,erR¨aumederenalshalbgenaueinfacdie-he
WgebnisurzelnbesagtdenganzenKorollarRaum5.2.4,dassaufspannen.dAls-dimensionaleweiteresreflexivkonvePolytopex-geometrisceh¨ocheshstensEr-
2dFacettenmitWurzelnimInnerenbesitzen,wobeiGleichheitnurf¨ur[−1,1]d

ZusammenfassungindeutscherSprache179
gilt.DasersteHauptresultatdiesesKapitels,Satz5.1.25,lieferteineErkl¨arung
f¨urBeobachtungeninderDatenbank¨uberdieAnzahlanWurzelnreflexiver
Polytope.Eswirdgezeigt,dassdiereduktiveAutomorphismengruppeeinerd-
dimensionalenvollst¨andigentorischenVariet¨at,diekeinProduktprojektiver
R¨aumeist,h¨ochstensDimension2f¨urd=2bzw.d2−2d+4f¨urd≥3besitzen
ann.kEineentscheidendeMotivation,sichmitderWurzelmengeauseinanderzuset-
zen,r¨uhrtvoneinemErgebnisvonBatyrevundSelivanovain[BS99]her,wel-
chesbesagt,dasseineglattetorischeFano-Variet¨ateineEinstein-K¨ahler-Metrik
besitzt,fallsdielinearenAutomorphismendesDualendesentsprechendenglat-
tenFano-Polytopskeinengemeinsamennicht-trivialenFixpunktbesitzen.Da
nacheinembekanntenSatzvonMatsushimadieExistenzeinerEinstein-K¨ahler-
gibtMetriksichalsimpliziert,unmittelbaresdassdieKorollar,AutomorphismengruppdassjedesereflexivdereVParietolytop,¨atreduktivdessenist,Dualeser-
einglattesFano-PolytopistunddessenlineareAutomorphismenkeinengemein-
samennicht-trivialenFixpunkthaben,nurhalbeinfacheWurzelnbesitzt.Die
sesAutorenResultat.fragtenDar¨ubdahererhinausnachveinemermutetereinkBatonvyrev,dassex-geometriscdieHalbheneinfacBeweishheitf¨urallerdie-
WurzelneinesreflexivenPolytopsschonausdemVerschwindenseinesSchwer-
punktesfolgt.ImebenbeschriebenenglattenFallistk¨urzlichvonWangundZhu
in[WZ04]sogargezeigtworden,dassdieseBedingung¨aquivalentzurExistenz
ist.ahler-Metrik¨Einstein-KeinerImzweitenHauptsatzdiesesKapitels,Satz5.3.1,gebenwirnuneineReihe
kombinatorischerKriterienan,diehinreichendoder¨aquivalentzurHalbeinfach-
heitallerWurzelneinervollst¨andigentorischenVariet¨atodereinesreflexiven
Polytopssind.DieseBedingungenbeinhaltenobigeVermutungen;dabeisind
s¨amtlicheBeweisereinkonvex-geometrisch.
Kapitel6besch¨aftigtsichmitzentralsymmetrischenreflexivenPolytopen.
EinHauptresultat,Satz6.5.1,besagt,dass[−1,1]dbisaufIsomorphiedasein-
ziged-dimensionalezentralsymmetrischereflexivePolytopmitderMaximalzahl
von3dGitterpunktenist.ImBeweiswirddabeibenutzt,dassaußer[−1,1]d
jedesd-dimensionalezentralsymmetrischereflexivePolytopwenigerals2dWur-
zelnhat.WasdiemaximaleAnzahlderEckenangeht,sok¨onnenwirinSatz
6.2.2flexivendiePolytopsallgemeinebewVermeisen.utungDiesimistFalleineeinesAnweinfacendungheneineszenweiterentralsymmetriscHauptergeb-henre-
nissesdiesesKapitels,Korollar6.3.3zuSatz6.3.1,n¨amlicheinervollst¨andigen
KlassifikationbeliebigdimensionalersimplizialerreflexiverPolytope,dieeinzen-
VtralsymmetriscerallgemeinerunghesPundaarVvonereinfacFhacettenungbvonesitzen.ResultatenMitvHilfeonEwdiesesaldimSatzes,glattenderundeine
vonexpliziteWirthimKlassifikkomationbinatoriscbiszuheinfacDimensionhstenfF¨unfalldurcdarstellt,hgef¨uhrt.wirdWineiterSatzzeigen6.3.12wir,die
dassinjederDimensiondeinsolchesPolytopdiegleichekombinatorischeStruk-
turwieimglattenFallhat,h¨ochstens2d2+1Gitterpunktebesitzt,in[−1,1]d
einbettbaristunddassdiesf¨urdasDualenochind/2[−1,1]dm¨oglichist.Ein
letztesHauptresultat,Satz6.4.4,gibteinemildeBedingungan,unterderein
allgemeinesd-dimensionalesreflexivesPolytopineinVielfachesvon[−1,1]dein-
gebettetwerdenkann.BesitztdemnacheinzentralsymmetrischesPolytopeine
einfacheFacette,soisteineEinbettunginein(2d−3)fachesm¨oglich.

AppendixB-Lebenslauf

Reutlingeninorengeb06.02.19761982–1986BesuchderGrundschuleinEntringen
1986–1995BesuchdesUhland-GymnasiumsT¨ubingen
1995AbituramUhland-GymnasiumT¨ubingen
1995–1996ZivildienstamInstitutf¨urmedizinische
InformationsverarbeitungT¨ubingen
1995–2001MathematikdiplomstudiummitNebenfachInformatik
anderUniversit¨atT¨ubingen
WiSe1999/2000AuslandsaufenthaltanderUniversityofSussex
imRahmenvon”InternationalStudiesinMathematics”
WiSe2000/2001Diplomarbeit¨uber”BerechnungdesEliminationsgrades”,
betreutvonProf.Dr.G.Scheja
WiSe2000/2001,WissenschaftlicheHilfskraftamMathematischenInstitut
SoSe2001derUniversit¨atT¨ubingen
Oktober2001DiplominMathematikanderUniversit¨atT¨ubingen,
BeginnderPromotion,betreutvonProf.Dr.V.V.Batyrev
seitOktober2001WissenschaftlicherAngestellterimArbeitsbereichAlgebra
amMathematischenInstitutderUniversit¨atT¨ubingen

ZumeinenakademischenLehrerninMathematikgeh¨orten:
G.Betsch,K.-J.Engel,C.Hering,F.Loose,G.Scheja,P.Schmid,M.Voit,
yrevBatV.V.

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