RATIONAL POINTS OF RATIONALLY SIMPLY CONNECTED VARIETIES
57 pages
English

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Niveau: Supérieur, Doctorat, Bac+8
RATIONAL POINTS OF RATIONALLY SIMPLY CONNECTED VARIETIES JASON MICHAEL STARR Abstract. These are notes prepared for a series of lectures at the conference Varietes rationnellement connexes: aspects geometriques et arithmetiques of the Societe Mathematique de France held in Strasbourg, France in May 2008. Contents 1. Introduction 1 Part 1. Rationally simply connected fibrations 7 2. The Kollar-Miyaoka-Mori conjecture 7 3. Sections, stable sections and Abel maps 8 4. Rational connectedness of fibers of Abel maps 15 5. The sequence of components 20 6. Rational connectedness of the boundary modulo the interior 26 7. Rational connectedness of the interior modulo the boundary 34 8. Rational simply connected fibrations over a surface 43 Part 2. Homogeneous spaces 45 9. Rational simple connectedness of homogeneous spaces 45 10. Discriminant avoidance 52 Part 3. The Period-Index Theorem and Serre's “Conjecture II” 53 11. Statement of de Jong's theorem and Serre's conjectures 53 12. Reductions of structure group 54 References 56 1. Introduction The goal of these notes is to present some new results proved jointly with A. J. de Jong and Xuhua He. First, an algebraic fibration over a surface has a rational section if the fiber is “rationally simply connected” and if the elementary obstruction vanishes. Second, this implies the split, geometric case of a conjecture of Serre, “Conjecture II” in [Ser02, p.

  • curve class

  • over

  • geometric generic

  • projective morphism

  • section curve

  • rationally connected

  • connected

  • there exists


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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

|
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

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