Gorenstein toric Fano varieties [Elektronische Ressource] / Benjamin Nill

Description

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 . . . . . . . . . . . . . . . . . . . . . . . . . .

Sujets

• Mathematics

Informations

• Publié par: eberhard_karls_universitat_tubingen
• Publié le: 01 janvier 2005
• Nombre de lectures: 59
• Langue: Deutsch
• Poids de l'ouvrage: 1 Mo

Extrait

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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
succ