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RATIONAL POINTS OF RATIONALLY SIMPLY CONNECTED VARIETIES

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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“porcupine”
todenoteareduciblecurveobtainedfroma(
g
)-freesectionbyattachingfreelines
infiberof
π
.Usingtheseporcupines,weproduceasequence(
Z
e
)
e


ofirreducible
components
Z
e
ofΣ
e
(
X/C/κ
).Ofcoursethispresupposestheexistenceofmany
freelinestoattachtoouroriginalsection,andthatleadstoourfirsttechnical
3

hypothesis:everypointofeverygeometricfiber
X
t
of
π
iscontainedinfreelinesin
X
t
,andeverylinein
X
t
isfree.Moreover,wewilldemandthattheparameterspace
forlinesin
X
t
containingafixedpointisitselfintegralandrationallyconnected.
Nowthesequence(
Z
e
)
e


isstillnotunique.Butifweassumethattheparam-
eterspaceforchainsoflinesin
X
t
containingtwofixed,generalpointsisalso
nonempty,integralandrationallyconnected,thenthesequenceis“asymptotically
unique”:foreveryotherchoiceofstartinginteger
e
=

0
andforeverysequence
W
0ofΣ

(
X/C/κ
)parameterizing(
g
)-freesections,thesequence(
Z
e
)
e


and(
W
e
)
e


0
becomeequalforall
e

0.ThisimpliesaGaloisinvariancepropertyforthe
sequence(
Z
e
)
e


.Thereforetoprovetheexistenceofasequence(
Z
e
)
e


ofcom-
ponents
Z
e
ofΣ
e
(
X/C/κ
)suchthat
α
|
Z
e
:
Z
e

Pic
e
(
X/C/κ
)
isdominantwithintegralandrationallyconnectedgeometricgenericfiber,itsuffices
toprovetheexistenceofasequenceofcomponents(
Z
e,κ
)
e


ofcomponents
Z
e,κ
of
thebase-changeΣ
e
(
X
κ
/C
κ

).Sowearereducedtoworkingoverthealgebraically
closedfield
k
(
t
).
Thehypothesesaboveimplythatthereexistsasequence(
Z
e
)
e


suchthateach
α
|
Z
e
isdominantwithintegralgeometricgenericfiber.Butweneedanadditional
hypothesistoprovethatthegeometricgenericfiberisrationallyconnected:the
existenceofa“2-twistingscroll”inthegeometricgenericfiberof
f
.Bycarefully
analyzinghowtheparameterspacesΣ
e
(
X/C/κ
)changeundertheporcupineop-
erationmentionedabove,weareabletoshowthatthesehypothesesdoimplythat
thegeometricgenericfiberof
α
|
Z
e
isrationallyconnectedforall
e

0.
Thisanalysisisquiteintricate.Infacttherearejustasmallnumberofsimple,geo-
metricideasinvolved.Butthereisalsoalargeamountofnotationandbookkeeping.
Wetriedtoatleastchoosememorablenamestoeasethenotation:“porcupines”,
“quills”oftheporcupine,“pens”tokeepporcupinestogether,etc.Stilltheamount
ofnotationandthelargenumberofsmall,technicallemmasbothremainserious
obstaclestounderstandingthemainarguments.TohelpIhaveaddedquiteabit
of“discussion”toSections6and7.
Aftertheproofofthemaintheorem,thereisstilltheissueofverifyingthehy-
pothesesforthevarietiesrelevanttoSerre’s“ConjectureII”:projectivehomoge-
neousspacesforsemisimplealgebraicgroups.Somehypothesesarestraightforward
toverify,andweexplainthesecompletelyinPart2.Butthemainhypothesis,
existenceofa2-twistingscroll,isanontrivialresultduetoourcoauthorXuhua
He.He’sproofiselegantandeasytofollow.Butitinvolvesasubstantialfraction
ofthetheoryofrootsystems,sowehavechosentoleaveHe’stheoremasa“black
.”xobFinallyinPart3weexplainhowSerre’s“ConjectureII”,aswellasdeJong’s
“Period-IndexTheorem”,eachreducetoexistenceofsectionsoffibrationsover
surfaceswhosegeometricgenericfiberisahomogeneousspace.
Indexofsomefrequentnotations.
(1)
f
:
X

S
.SeeCorollary8.1.
f
isaprojective,flatmorphismtoasurface
S
overanalgebraicallyclosedfield.
4

(2)
X/C/κ
and
L
,
g
,
K
and
Y
.SeeNotation2.4.
X/C/κ
denotesaprojective,
flatmorphism
π
:
X

C
where
C
isa
κ
-curveofgenus
g
.
L
isa
π
-ample
invertiblesheafon
X
.
K
isthealgebraicclosure
K
=
κ
(
C
).
Y
isthe
geometricgenericfiberof
π
.
e(3)Pic
C/κ
.ThePicardschemeparameterizingfamiliesofdegree
e
invertible
sheaveson
C
.
(4)
δ
;
t
1
,...,t
δ
.SeethediscussionfollowingDefinition3.2.
δ
isanonnegative
integerand
t
1
,...,t
δ
arepointsof
C
.
(5)
P

Q
.Thisisnotationforaquasi-projectivemorphism,particularly
whendescribinggeneralconstructionswhichapplytoanyquasi-projective
morphism.
e(6)Sec(
X/C/κ
)and
σ
.SeeTheorem3.1.Theuniversalspaceofsectionsof
π
:
X

C
whichhavedegree
e
withrespectto
L
.
(7)
α
.SeeSection3.
α
alwaysdenotesanAbelmapwhosetargetisPic
eC/κ
for
someinteger
e
.
(8)
γ
,
n
,
β
.SeeSection3.
γ
isthearithmeticgenusofanodalcurve,
n
is
thenumberofmarkedpointsonthecurve,and
β
isacurveclassonsome
quasi-projectivescheme
P
,i.e.,ahomomorphismPic(
P
)

Z
.
(9)
M
γ,n
(
P,β
)andM
γ,n
(
P,β
).SeeSection3;
M
γ,n
(
P,β
)isaDeligne-Mumford
stackparameterizinggenus
γ
,
n
-pointedstablemapsto
P
withcurveclass
β
,andM
γ,n
(
P,β
)isthecoarsemodulispaceof
M
γ,n
(
P,β
).
(10)ev
γ,n,β
:M
γ,n
(
P,β
)

P
n
.SeeSection3.Theevaluationmapassociating
toastablemapfroman
n
-pointedcurveto
P
the
n
-tupleofimagesofthe
n
markedpointsin
P
.
(11)Σ
e
(
X/C/κ
).SeeDefinition3.2.Thecoarsemodulispaceofstableporcu-
pinesof
X/C/κ
whichhavedegree
e
withrespectto
L
;thisisprojectiveand
econtainsthequasi-projectiveschemeSec(
X/C/κ
)asanopensubscheme.
(12)
h
:
C
0

X
,
σ
0
:
C

X
,
D
=
t
1
+
∙∙∙
+
t
δ
,and
C
1
0
,...,C
δ
0
.Seethe
discussionfollowingDefinition3.2.
h
isastablesection,
σ
0
istheunique
sectionsuchthat
σ
0
(
C
)iscontainedin
h
(
C
0
),
D
isthedivisorofnodesof
h
(
C
0
)containedin
σ
0
(
C
),and
C
i
0
isthemaximalverticalsubcurveof
C
0
whoseimageunder
h
contains
σ
0
(
t
i
).
(13)
Z
e
.SeeDefinition4.8andDefinition5.6.
Z
e
isanirreduciblecomponent
ofΣ
e
(
X/C/κ
).
(14)M
γ,n
(
P/Q,e
).SeeDefinition3.4.Thespaceofstablemapsrelativetoa
morphism
f
:
P

Q
,i.e.,stablemapsto
P
whoseimageiscontainedin
afiberof
f
andwhichhavedegree
e
withrespecttoan
f
-ampleinvertible
sheaf.
(15)
f
γ,n,β
:M
γ,n
(
P/Q,e
)

Q
.SeethediscussionfollowingDefinition3.4.
Themapassociatingtoastablemapintoafiberof
f
thepointof
Q
which
istheimageofthestablemapunder
f
.
(16)ev
γ,n,β
:M
γ,n
(
P/Q,e
)

P
×
Q
∙∙∙×
Q
P
.Seethediscussionfollowing
Definition3.4.Theevaluationmapasinitem10.
(17)
R
and
ρ
.SeeDefinition3.5.
R
isa
scroll
1
for
X/C/κ
,i.e.,aclosed
subscheme
R
of
X
suchthattheprojection
π
|
R
:
R

C
issmoothand
surjectivewithgeometricfibersbeinglines.Themorphism
π
|
R
isusually
denoteby
ρ
.
1
scroll
Engl.=
surfacere´gle´e
Fr.
5

(18)(
R,L
).SeeDefinition3.6.An
m
-twistingscrollfor
X/C/κ
,i.e.,ascroll
R
for
X/C/κ
togetherwithaCartierdivisorclass
L
on
R
satisfyingvarious
properties.
(19)Chn
2
(
P/Q,n
).SeeDefinition3.7.Themodulispaceof2-pointedchains
(
C
0
,p
1
,q
n
)=((
L
1
,p
1
,q
1
)
,...,
(
L
n
,p
n
,q
n
))of
n
linesinfibersof
P

Q
.
(20)ev
0
,
2
,n
:Chn
2
(
P/Q,n
)

P
×
Q
P
.Theevaluationmap,seeitem16.
(21)Φ:M
0
,
1
(
X/C,
1)

M
0
,
0
(
X/C,
1).SeeSection4.Theforgetfulmorphism
associatingtoapointedline(
L,p
)theline
L
.
(22)(
g
)asin“(
g
)-free”.SeeDefinition4.7.(
g
)isthemaximumof1and2
g

1.
(23)(
Z
e
)
e


.SeeDefinition4.8.Asequenceforallinteger
e


ofanirreducible
component
Z
e
ofΣ
e
(
X/C/κ
).
(24)Porc
e,δ
(
X/C/κ
).SeeDefinition5.1andProposition5.2.Theparameter
spacefor
porcupines
2
.Aporcupineisaspecialstablemap
h
:
C
0

X
.The
sectioncomponent
C
0
isthe
body
3
.Theverticalcomponentsarethe
quills
4
.
Theinteger
e
denotesthetotaldegreeoftheporcupineand
δ
denotesthe
numberofquills.
(25)
T
P/Q
or
T
f
.The
verticaltangentbundle
associatedtoamorphism
f
:
P

Q
,i.e.,thedualofthesheafofrelativedifferentialsof
f
(usuallyonly
appliedwhenthesheafofrelativedifferentialsislocallyfree).
(26)Φ
body
:Porc
e,δ
(
X/C/κ
)

Porc
e

δ,
0
(
X/C/κ
).SeeLemma5.3.Themor-
phismassociatingtoeachporcupine
h
:
C
0

X
thebody
σ
o
:
C

X
of
theporcupine.
(27)Φ
0
body
:Porc
e,δ
(
X/C/κ
)

Porc
e

δ,
0
(
X/C/κ
)
×
κ
C
δ
.SeeLemma5.3.
Themorphismassociatingtoeachporcupine
h
:
C
0

X
the
extended
body
(
σ
0
,D
),i.e.,thebody
σ
0
togetherwiththeattachmentdivisor
D
=
t
1
+
∙∙∙
+
t
δ
,seeitem12.
(28)
X
(
t
1
,...,t
δ
)
.SeeNotation6.1.Givendistinctclosedpoints
t
1
,...,t
δ
of
C
,
notationforthefiberproductofthecorrespondingfibersof
π
,
X
t
1
×
κ
∙∙∙×
κ
.Xtδ(29)Porc
e,δ
(
X/C/κ
)
Z
.Givenasequence(
Z
e
)
e


asinitem23,theintersection
ofPorc
e,δ
(
X/C/κ
)and
Z
e
insideΣ
e
(
X/C/κ
).
(30)Chn
2
(
X/C,n
)
(
t
1
,...,t
δ
)
.SeeNotation6.1.
t
1
,...,t
δ
isasinitem28.And
Chn
2
(
X/C/κ,n
)
(
t
1
,...,t
δ
)
isthefiberproductChn
2
(
X
t
1
/κ,n
)
×
κ
∙∙∙×
κ
Chn
2
(
X
t
δ
/κ,n
).
O(31)
O
andChn
2
(
X/C,n
)
(
t
1
,...,t
δ
)
.SeeNotation6.1.
O
isadenseopensub-
Osetof
X
(
t
1
,...,t
δ
)
.AndChn
2
(
X/C,n
)
(
t
1
,...,t
δ
)
istheopensubschemeof
Chn
2
(
X/C,n
)
(
t
1
,...,t
δ
)
parameterizingchainssuchthateveryassociatedse-
quenceofmarkedpointsornodesin
X
(
t
1
,...,t
δ
)
iscontainedinthedense
open
O
.
(32)
O
C
(Γ).Aninvertiblesheafon
C
,especiallyingeneralconstructionsthat
applytoanyinvertiblesheafon
C
.
(33)
ζ
:
C

M
0
,
1
(
X/C,
1)or
τ
:
C

M
0
,
1
(
X/C,
1),
R
and
σ
0
.SeeDefinition
7.2andLemma7.3.
ζ
isasectionoftheprojection
π
0
,
1
,
1
:M
0
,
1
(
X/C,
1)

C
,seeitem15,andsometimessois
τ
whenmorethanonesuchsection
needstobeused.Asectionof
π
0
,
1
,
1
determinesascroll
R
,seeitem17,
2
porcupine
Engl.=
porc-e´pic
Fr.
3
body
Engl.=
corps
Fr.
4
quill
Engl.=
piquant
Fr.

6

togetherwithasection
σ
0
of
ρ
:
R

C
.Thesection(ormoregenerally
aporcupinewhosebodyequalsthesection)issaidtobe
penned
5
bythe
scroll
R
,and
R
iscalleda
pen
6
forthesection.
(34)
N
P/Q
or
N
i
.Thenormalsheafofaclosedimmersion
i
:
P

Q
.
(35)Porc
e,δ
(
X/C/κ
).SeeProposition7.8.TheclosureinΣ
e
(
X/C/κ
)ofthe
locallyclosedsubschemePorc
e,δ
(
X/C/κ
).
(36)
α

1
(
O
C
(Γ)).ThefiberoftheAbelmap
α
overapoint
O
C
(Γ)ofPic
C/κ
.
(37)
G
,
P
,
Q
and
R
,
B
,
R
u
(
B
),
T
,Φ,
I
.SeeSection9.
G
isasemisimple
algebraicgroup.
P
,
Q
and
R
areallparabolicsubgroupsof
G
.
B
isaBorel
subgroupof
G
.
R
u
(
B
)istheunipotentradicalof
B
.
T
isamaximaltorus
of
G
.Φisarootsystemand
I
isthesetofsimpleroots.
(38)
G
/T
,
T
/T
,
X
T
.SeeSection10.
T
isascheme.
G
isareductivegroup
schemeover
T
.And
T
isa
G
-torsorover
T
.Givena
T
-scheme
X
together
withanactionof
G
as
T
-schemes,
X
T
isthecorrespondingtwistof
X
.
(39)Br(
K
).SeeSection11.Br(
K
)istheBrauergroupofafield
K
.
Acknowledgments.
Manypeopledeservethanksfortheirvaluablecontributions
tothisproject.ThepersonmostdeservingofthanksisPhilippeGillewhobasically
showedushowtoproveTheorem12.1,assumingtheresultsinParts1and2.
TheeditorJean-LouisColliot-The´le`nealsodeservesspecialthanksforhismany
suggestionsandforhisremarkablepatience.
Part
1.
Rationallysimplyconnectedfibrations
2.
TheKolla´r-Miyaoka-Moriconjecture
Let
k
beanalgebraicallyclosedfield.Asmooth,connected
k
-scheme
X
is
rationally
connected
,resp.
separablyrationallyconnected
,ifthereexistsanintegral
k
-scheme
M
anda
k
-morphism
h
:
M
×
k
P
k
1

X,
(
m,t
)
7→
h
(
m,t
)
suchthattheinducedmorphism
h
0
,

:
M

X
×
k
X,m
7→
(
h
(
m,
0)
,h
(
m,

))
isdominant,resp.dominantandseparable.Roughlythissaysthateverypairof
pointsin
X
iscontainedintheimageofamorphismwithdomain
P
k
1
,i.e.,every
pairisconnectedbyarationalcurve.Moregenerallyforanintegral
k
-scheme
X
whichispossiblysingular,wewillsometimessaythat
X
is
rationallyconnected
,
resp.
separablyrationallyconnected
,ifthereexistsaprojectivebirationalmorphism
X
˜

X
suchthatthesmoothlocusof
X
˜isrationallyconnected,resp.separably
rationallyconnected.
Amongsmooth,projectivevarieties,everyunirationalvarietyisrationallycon-
nected.Theconverseisunknown,butitisexpectedtobefalse.Rationalconnect-
ednesssatisfiesmanynicepropertieswhichareunknownforunirationality.Oneof
theseproperties,whichFanoconjecturedfailsforunirationality,isthatthetotal
spaceofanalgebraicfibrationofsmooth,projectivevarietiesisseparablyrationally
connectedifthebaseandgeneralfiberarebothseparablyrationallyconnected
5
penned
Engl.=
enclos
Fr.
6
pen
Engl.=
enclos
Fr.

7

(therearecounterexamplesshowingonecannotreplace“separablyrationallycon-
nected”by“rationallyconnected”).Thisfollowsfromastrongerresult,originally
conjecturedbyKolla´r-Miyaoka-Mori[Kol96,ConjectureIV.6.1.1].
Theorem2.1.
[GHS03]
,
[dJS03]
Let
k
beanalgebraicallyclosedfield.Let
C
bea
smooth,projective
k
-curve.Let
π
:
X

C
beaprojective,flatmorphism.Assume
thatthegeometricgenericfiber
Y
of
π
isnormalandthatthesmoothlocusof
Y
isseparablyrationallyconnected.Thenthereexistsa
k
-morphism
σ
:
C

X
such
that
π

σ
=
Id
C
,i.e.,
σ
isasectionof
π
.
Thishasanumberofconsequenceswhicharediscussedelsewhere,cf.[Deb03].
Twoofthese,whichKolla´r-Miyaoka-Morideducedfromtheirconjecture,arequite
usefulinwhatcomeslater.
Corollary2.2.
[KMM92]
Withhypothesesasabove,let
t
1
,...,t
δ
bedistinct
k
-
pointsof
C
suchthatthefiber
X
t
j
of
π
issmoothfor
j
=1
,...,δ
.Foreach
j
=1
,...,δ
,let
p
j
bea
k
-pointof
X
t
j
.Thereexistsasection
σ
:
C

X
of
π
such
that
σ
(
t
j
)
equals
p
j
forevery
j
=1
,...,δ
.
Corollary2.3.
[KMM92]
Assumethatchar
(
k
)
equals
0
.Let
f
:
P

Q
bea
surjectivemorphismofintegral,projective
k
-schemes.Assumethat
Q
isrationally
connectedandassumethatthegeometricgenericfiberof
f
isintegralandrationally
connected.Thenalso
P
isrationallyconnected.
Assumenowthatchar(
k
)equals0.Byresolutionofsingularitieswemayassume
X
issmooth.Andbygenericsmoothnessthereareatmostfinitelymanysingular
fibersof
π
.Thus,byCorollary2.2,
π
has
many
sections.Inparticular,exceptin
thetrivialcasethat
π
isanisomorphism,the
k
-schemeparameterizingsectionsof
π
hascomponentswhosedimensionsbecomearbitrarilylarge.Underadditionalhy-
potheseson
π
wecanprovethatthesecomponents“eventuallybecomeasrationally
connectedaspossible”.Theprecisestatementisgivenbelow.
Fortheapplicationsitisimportanttoalsoconsiderthecasewhenthegroundfield
isnotalgebraicallyclosed.
Notation2.4.
Let
κ
denoteacharacteristic0field(possiblynotalgebraically
closed).Let
C
denoteasmooth,projective,geometricallyconnected
κ
-curve.De-
notethegenusof
C
by
g
.Let
X
denoteasmooth,projective
κ
-scheme.Let
π
:
X

C
denoteasurjectivemorphismsof
κ
-schemeswhosegeometricgeneric
fiberisirreducible.Denoteby
K
thealgebraicallyclosedfield
K
=
κ
(
C
),andde-
noteby
Y
thegeometricgenericfiberof
π
whichisasmooth,projective,connected
K
-scheme.Finally,let
L
denoteaninvertiblesheafon
X
whichis
π
-ample.And
denoteby
L
Y
thebase-changeof
L
to
Y
.
3.
Sections,stablesectionsandAbelmaps
Theproofofthemaintheoremusesseveraldifferentkindsofcurvesin
X
,some
ofwhicharereducible.Thissectionintroducesthesedifferenttypesofcurvesand
reviewsthebasicsoftheparameterspacesforthesecurves.Ofcoursethemost
importantcurvesaresectioncurves:imagesofsectionsof
π
.Buttheparameter
spacesforsuchcurvesarenotproper.Theyareopensubsetsofpropermoduli
spaces,themodulispacesof“stablesections”.Themainresultofthissectionis
8

thattheAbelmapextendstothemodulispaceofstablesectionsandthattheimage
ofastablesectionundertheAbelmapcanbeunderstoodbyasimpleanalysisof
thecomponentsofthestablesection.
ThenotationshereareasinNotation2.4.Let
S
bea
κ
-scheme.A
familyof
sectionsof
π
:
X

C
parameterizedby
S
isamorphismof
C
-schemes
τ
:
S
×
κ
C

X.
Foraninteger
e
,thefamilyofsectionshas
degree
e
iftheinvertiblesheaf
τ

L
on
S
×
κ
C
hasrelativedegree
e
over
S
,i.e.,foreverygeometricpoint
s
of
S
thebase-
changeof
τ

L
to
C
s
hasdegree
e
.Apair(
S,τ
)asaboveis
universal
ifforevery
κ
-scheme
S
0
andforeveryfamilyofdegree
e
sectionsof
π
parameterizedby
S
,
τ
0
:
S
0
×
κ
C

X,
thereexistsaunique
κ
-morphism
f
:
S
0

S
suchthat
τ
0
equals
τ

(
f,
Id
C
).
Theorem3.1
(Grothendieck)
.
[Gro62,PartIV.4.c,p.221-19]
Foreveryinteger
e
thereexistsa
universalpair(
Sec
e
(
X/C/κ
)

)
ofa
κ
-schemeandafamilyofdegree
ee
sectionsof
π
parameterizedbySec
(
X/C/κ
)
,

:
Sec
(
X/C/κ
)
×
κ
C

X.
eMoreoverSec
(
X/C/κ
)
isaquasi-projective
κ
-scheme.
eInvertiblesheaveson
C
ofdegree
e
areparameterizedbythePicardschemePic
C/κ
.
Thus,associatedtotheinvertiblesheaf
σ

L
thereisamorphismof
κ
-schemes
eeα
:Sec(
X/C/κ
)

Pic
C/κ
.
Thismorphismisthe
Abelmap
associatedto
L
.
Stablemaps,stabilizationandevaluationmorphisms.
The
κ
-schemeSec
e
(
X/C/κ
)
isquasi-projective.Itisrarelyprojective.Itisconvenienttoworkwithaprojec-
tivescheme.FortunatelythereexistsaprojectiveschemeΣ
e
(
X/C/κ
)containing
Sec
e
(
X/C/κ
)asanopensubscheme:thecoarsemoduli(algebraic)spacefor“sta-
blesections”.Thiscomesfromamorefundamentalscheme:thecoarsemoduli
spacefor“stablemaps”.Anexcellentreferenceforstablemapsisthearticleof
FultonandPandharipande,cf.[FP97].Hereisaverybriefsummary.Forevery
quasi-projective
κ
-scheme
P
,fornonnegativeintegers
γ,n
,andforagrouphomo-
morphism
β
:Pic(
P
)

Z
,L
7→h
L,β
i
,
hereafterknownasa“curveclass”,a
familyof
n
-pointed,genus
γ
,stablemapsto
P
ofclass
β
parameterizedbya
κ
-scheme
S
isadatum
(
ρ
:
C→
S,
(
τ
i
:
S
→C
)
i
=1
,...,n
,h
:
C→
P
)
ofaproper,flatmorphism
ρ
:
C→
S
,
n
sectionmorphisms
τ
i
:
S
→C
of
ρ
:
C→
S
,
anda
κ
-morphism
h
:
C→
P
suchthatforeverygeometricpoint
s
of
S
,
(i)thefiber
C
s
of
ρ
over
s
isaconnected,at-worst-nodalcurveofgenus
γ
,
(ii)theimages
τ
i
(
s
)arepairwisedistinctandallcontainedinthesmoothlocus
,Cfos(iii)theinducedhomomorphism
∗Pic(
P
)

Z
,
L7→
deg
C
s
(
h
L|
C
s
)
equals
β
,and
9

(iv)thegroupofautomorphismsof
C
s
fixingeachpoint
τ
i
(
s
)andcommuting
with
h
|
C
s
isfinite.
Suchfamiliesaretheobjectsofacategory
M
γ,n
(
P,β
)whichisfiberedingroupoids
overthecategoryof
κ
-schemes
S
:themorphismsin
M
γ,n
(
P,β
)aswellasthe
clivage(i.e.,the“pullbacks”)aretheevidentones,cf.[Kon95].Thecategory
M
γ,n
(
P,β
)isaseparated,finitetypeDeligne-Mumfordstackover
κ
whichisproper
if
P
isproper,cf.[Kon95].Thecoarsemoduli(algebraic)space,M
γ,n
(
P,β
),ofthe
Deligne-Mumfordstack
M
γ,n
(
P,β
)isaquasi-projective
κ
-schemewhichisproper
if
P
isproper,cf.[FP97].
“Stabilization”isanotherusefulfeatureofstablemaps.Assumethatthetriple
(
γ,n,β
)is“stable”inthesensethateither
β
isnonzeroonsomeampleinvertible
sheaforelse(
γ,n
)isdifferentfrom(0
,
0),(0
,
1),(0
,
2),and(1
,
0).Foreveryfamily
ofmapssatisfying(i),(ii)and(iii)butnotnecessarily(iv)(suchfamiliesarecalled
“prestable”),thereexistsastablefamily
(
ρ
stab
:
C
stab

S,
(
τ
i,
stab
:
S
→C
stab
)
i
=1
,...,n
,h
stab
:
C
stab

P
)
andaproper,surjectivemorphism
u
:
C→C
stab
compatiblewith
h
andthemaps
τ
i
(i.e.,
h
equals
h
stab

u
andeach
τ
i,
stab
equals
u

τ
i
)andsuchthatforevery
geometricpoint
s
of
S
andeveryconnected,closedsubcurve
B
of
C
s
bothofthe
followinghold.
(i)Thesubcurve
B
iscontracted(toapoint)by
u
ifandonlyifitisan
unstabletree
:thearithmeticgenusof
B
equals0,
B
iscontractedby
h
,and
B
contains
<
3specialpoints,i.e.,markedpoints
τ
i
(
s
)andintersections
pointsof
D
with
C
s

D
.
(ii)If
B
containsnosubcurvewhichisanunstabletreethen
u
|
B
:
B

u
(
B
)is
abirational,unramifiedmorphismwhichidentifiesapairofdistinctpoints
of
B
ifandonlyifthepointsarecontainedinacommonunstabletreeof
.CsThestablefamilytogetherwiththepropermorphism
u
isa
stabilization
ofthe
originalprestablefamily.Thestabilizationisuniqueuptouniqueisomorphism,
andeverystablefamilyisitsownstabilization.Stabilizationiscompatiblewith
pullbacks.Themainapplicationofstabilizationisthefollowing.Let
f
:
P

Q
be
a
κ
-morphismofquasi-projective
κ
-schemes.Foreachfamilyofstablemapsto
P
(
ρ
:
C→
S,
(
τ
i
:
S
→C
)
i
=1
,...,n
,h
:
C→
P
)
thereisaninducedfamilyofprestablemapsto
Q
,
(
ρ
:
C→
S,
(
τ
i
:
S
→C
)
i
=1
,...,n
,f

h
:
C→
Q
)
.
Soif
f

β
:=
β

f

isnonzeroonsomeampleinvertible
O
Q
-moduleorif(
γ,n
)
isdifferentfrom(0
,
0),(0
,
1),(0
,
2)and(1
,
0)thenstabilizationdeterminesa1-
morphismofDeligne-Mumfordstacks
f

:
M
γ,n
(
P,β
)
→M
γ,n
(
Q,f

β
)
whichinturndeterminesamorphismofcoarsemodulispaces
f

:M
γ,n
(
P,β
)

M
γ,n
(
Q,f

β
)
.
Itissometimesalsousefulthatthemorphism
u
is
rational
,i.e.,
O
C
stab

u

O
C
is
anisomorphismand
R
q
u

O
C
iszeroforall
q>
0.
01