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Publié par | profil-zyak-2012 |
Nombre de lectures | 25 |
Langue | English |
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RATIONALPOINTSOFRATIONALLYSIMPLYCONNECTED
VARIETIES
JASONMICHAELSTARR
Abstract.
Thesearenotespreparedforaseriesoflecturesattheconference
Varie´te´srationnellementconnexes:aspectsge´ome´triquesetarithme´tiques
of
theSocie´te´Mathe´matiquedeFranceheldinStrasbourg,FranceinMay2008.
Contents
1.Introduction1
Part1.Rationallysimplyconnectedfibrations
7
2.TheKolla´r-Miyaoka-Moriconjecture7
3.Sections,stablesectionsandAbelmaps8
4.RationalconnectednessoffibersofAbelmaps15
5.Thesequenceofcomponents20
6.Rationalconnectednessoftheboundarymodulotheinterior26
7.Rationalconnectednessoftheinteriormodulotheboundary34
8.Rationalsimplyconnectedfibrationsoverasurface43
Part2.Homogeneousspaces
45
9.Rationalsimpleconnectednessofhomogeneousspaces45
10.Discriminantavoidance52
Part3.ThePeriod-IndexTheoremandSerre’s“ConjectureII”
53
11.StatementofdeJong’stheoremandSerre’sconjectures53
12.Reductionsofstructuregroup54
References56
1.
Introduction
ThegoalofthesenotesistopresentsomenewresultsprovedjointlywithA.J.
deJongandXuhuaHe.First,analgebraicfibrationoverasurfacehasarational
sectionifthefiberis“rationallysimplyconnected”andifthe
elementaryobstruction
vanishes.Second,thisimpliesthesplit,geometriccaseofaconjectureofSerre,
“ConjectureII”in[Ser02,p.137]:foraconnected,simplyconnected,semisimple
algebraicgroup,everyprincipalbundleforthegroupoverasurfacehasarational
section.ManyothershaveworkedtowardstheresolutionofSerre’s“ConjectureII”
inthegeometriccaseandinthegeneralcase:MerkurjevandSuslin;E.Bayerand
Date
:January3,2010.
1
R.Parimala;Chernousov;andP.Gille.Theseresultsaresummarizedin[CTGP04,
Theorem1.2(v)].Becauseofthesemanyresults,thefull“ConjectureII”inthe
geometriccasereducestothesplit,geometriccase,sothat“ConjectureII”isnow
settledinthegeometriccase.
Thesenotescloselyfollowourarticle[dJHS08].Buttheargumentsherearea
bitsimpler,andthehypothesesareconsiderablystronger(yetstillverifiedinthe
applicationtoSerre’sconjecture).
Thesenotesaccompanylecturesdeliveredattheconference
Varie´te´srationnelle-
mentconnexes:aspectsge´ome´triquesetarithme´tiques
oftheSocie´te´Mathe´matique
deFranceheldinStrasbourg,FranceinMay2008.Inadditiontothenewresults,
thelecturesalsopresentedtheproofoftheKolla´r-Miyaoka-Moriconjectureproved
byTomGraber,JoeHarrisandtheauthorincharacteristic0andbyA.J.de
Jongandtheauthorinarbitrarycharacteristic.Butastherearealreadyseveral
expositionsofthatwork,Iwillonlyreviewthemainstatement.
Overviewoftheproof.
Givenasmooth,projectivesurface
S
overanalge-
braicallyclosedfield
k
,therealwaysexistsaLefschetzpencilofdivisorson
S
.The
genericfiber
C
ofthispencilisasmooth,projective,geometricallyintegralcurve
overthefunctionfield
κ
=
k
(
t
).Givenaprojective,flatmorphism
f
:
X
→
S
whosegeometricgenericfiberisintegralandrationallyconnected,thefiberprod-
uct
X
κ
:=
C
×
S
X
isaprojective
κ
-schemetogetherwithaprojective,flatmorphism
of
κ
-schemes
π
:
X
κ
→
C
whosegeometricgenericfiberisintegralandrationally
connected.Sincethegenericof
π
equalsthegenericfiberof
f
,rationalsectionsof
f
arereallythesameasrationalsectionsof
π
.Soitsufficestoprovethat
π
hasa
section.
Andthemorphism
π
hasoneadvantageover
f
:thebasechangemorphism
π
⊗
Id:
X
κ
⊗
κ
κ
→
C
⊗
κ
κ
doeshaveasectionbyTheorem2.1.ByGrothendieck’sworkontheHilbertscheme
thereexistsa
κ
-schemeSections(
X/C/κ
)parameterizingfamiliesofsectionsof
π
.
ThegoalistoproveSections(
X/C/κ
)hasa
κ
-point,butweatleastknowithas
a
κ
-point.AswithallHilbertschemes,thisisreallyacountableunionofquasi-
eeprojective
κ
-schemes,
t
e
Sections(
X/C/κ
),whereSections(
X/C/κ
)istheopen
andclosedsubschemeparameterizingsectionswhichhavedegree
e
withrespectto
some
π
-relativelyampleinvertiblesheaf
L
.
ThebasicideaistotrytoprovethatSections
e
(
X/C/κ
)hassomenaturallydefined
closed
κ
-subschemewhichisgeometricallyintegralandgeometricallyrationally
connected.ThenwecanapplyTheorem2.1tothisclosedsubschemetoproducea
κ
-pointofSections
e
(
X/C/κ
),whichisthesameasasectionof
π
.
OfcoursethereisanobstructiontorationalconnectednessofSections
e
(
X/C/κ
):
theAbelmap
α
:Sections
e
(
X/C/κ
)
→
Pic
eC/κ
sendingeachsectionof
π
tothepullbackof
L
bythissection.Sincethereareno
rationalcurvesintheAbelianvarietyPic
eC/κ
,everyrationallyconnectedsubvariety
ofSections
e
(
X/C/κ
)iscontainedinafiberof
α
.Sotheideaistoprovethat
for
e
sufficientlypositive,someirreduciblecomponentofthegenericfiberof
α
is
geometricallyintegralandgeometricallyrationallyconnected.Ofcoursethisisthe
2
sameasprovingthatthereexistsanirreduciblecomponent
Z
e
ofSections
e
(
X/C/κ
)
suchthat
eα
|
Z
e
:
Z
e
→
Pic
C/κ
isdominantwithintegralandrationallyconnectedgeometricgenericfiber.Observe
thatthiswouldbeenoughtoconcludetheexistenceofasectionof
π
:thereare
κ
-pointsofPic
eC/κ
,e.g.,comingfromthebasepointsoftheLefschetzpencil,and
thefiberof
α
|
Z
e
overthese
κ
-pointsisthenageometricallyintegralandrationally
connectedvarietydefinedover
κ
=
k
(
t
).Suchavarietyhasa
κ
-pointbyTheorem
.1.2Therearesomeissues.Firstofallifwechange
L
thentheAbelmap
α
changes.
Forinstance,ifwereplace
L
by
L
⊗
n
with
n>
1,thentheoriginalAbelmapis
composedwiththe“multiplicationby
n
”morphismonthePicardscheme.Because
thisisafinitemapofdegree
>
1,thegeometricgenericfiberofthenewAbelmap
willnotbeintegral.Soitiscrucialtoworkwiththecorrectinvertiblesheaf
L
.
Ifthegeometricgenericfiberof
f
hasPicardgroupisomorphicto
Z
(rationally
connectedvarietiesalwayshavediscretePicardgroup),thenthisobstructionis
equivalenttothewellknown
elementaryobstruction
ofColliot-The´le`neandSansuc.
Weimposevanishingoftheelementaryobstructioninasomewhathiddenmanner
throughexistencepropertiesfor“lines”inthegenericfiber,i.e.,curvesof
L
-degree
1.Observethattherearenocurvesof
L
⊗
n
-degree1,whichindicatestheconnection
withtheelementaryobstruction.
Asecond,weightierissueisthatSections
e
(
X/C/κ
)typicallyisnotproper.Soit
isextremelyunlikelyanyinterestingsubvarietiesarerationallyconnected.Fortu-
natelyitsufficestoprovethereisacomponent
Z
e
asaboveforacompactification
Σ
e
(
X/C/κ
)ofSections
e
(
X/C/κ
).Thecompactificationweuseherecomesfrom
Kontsevich’smodulispaceofstablemaps.Butthereisathirdproblem:thisspace
willusuallyhavemorethanoneirreduciblecomponent.Someofthesecomponents
havebadpropertiesbecausethegenericpointparameterizesanobstructedsection.
Sowerestrictattentiontothoseirreduciblecomponentswhichparameterizeunob-
structedsections,specificallywhatwecall“(
g
)-freesections”where
g
isthegenus
of
C
.Stilltheremaybemorethanoneirreduciblecomponent
Z
parameterizing
(
g
)-freesections.
Wecannotfixthisforanyparticularinteger
e
:foranyparticularinteger
e
=
there
maywellbemorethanoneirreduciblecomponent
Z
ofΣ
(
X/C/κ
)parameterizing
(
g
)-freesections.Howevertheproblemgetsbetteras
e
becomesmorepositive.
Thereisastandardwayofproducingnewsectionsfromold:attachverticalrational
curvestothesectioncurveanddeformthisreduciblecurvetogetanirreducible
curvewhichisagainasection.Iftheoriginalsectioncurveandverticalcurvesare
sufficientlyfree,thenthereduciblecurvedoesdeformandthedeformationsare
againunobstructed.Inparticularthenewsectionisparameterizedbyasmooth
0pointofΣ
e
(
X/C/κ
)forsome
e
0
>e
.Ofcoursetherearemanywaysofattaching
anddeforming,sowechoosethesimplestpossible:attachvertical“lines”,i.e.,
curveswhose
L
-degreeequals1.Weusethesomewhatcolorfulname“porcupine̶