Generalized theta linear series on moduli spaces of vector bundles on curves

33 Pages·0.36 MB·English

by  Mihnea Popa

GENERALIZED THETA LINEAR SERIES ON MODULI SPACES OF VECTOR
BUNDLES ON CURVES
0
1 MIHNEAPOPA
0
2

n
a
CONTENTS
J

7 1. Introduction 1
1 2. Semistablebundles 2

2.1. Arbitraryvectorbundles 2
]
G 2.2. Semistablevectorbundles 4
2.3. Example:Lazarsfeld’sbundles 6
A
2.4. Example:Raynaud’sbundles 8
.
h 2.5. Themodulispace 9
t
a 3. Generalizedthetadivisors 11
m 4. Quotschemesandstablemaps 14
[ 5. VerlindeformulaandStrangeDuality 15
5.1. Verlindeformula 15
4
5.2. StrangeDuality 17
v
6. Basepoints 19
2
9 6.1. Abstractcriteria 19
1 6.2. Concreteconstructions 20
3 6.3. Otherdeterminantsorlevels 22
.
2 7. Effectiveresultsongeneralizedthetalinearseries 23
1 7.1. Thethetamap 23
7
7.2. Basepoint-freeness 24
0
7.3. Veryampleness 26
:
v 7.4. VerlindebundlesandboundsforU (r,d) 27
X
i
X 7.5. Conjectures 30
References 31
r
a
1. INTRODUCTION
This article is based on lecture notes prepared for the August 2006 Cologne Summer
School. Thenoteshaveexpandedinthemeanwhileintoasurveywhosemainfocusisthe
studyoflinearseriesonmodulispacesofvectorbundlesoncurves.
Thepaperisessentiallydividedintotwoparts.Thefirst,smallerone,consistsof§2and
§3,andisalmostentirelydevotedtobackgroundmaterial,reflectingpreciselywhatIwas
TheauthorwaspartiallysupportedbytheNSFgrantDMS-0500985,byanAMSCentennialFellowship,and
byaSloanFellowship,atvariousstagesinthepreparationofthiswork.
1
2 MihneaPopa
abletocoverduringthefirstcoupleofintroductorylecturesattheschool. Thereareafew
shortproofsorhints, someexercises, while forthefundamentalresultsI haveto referto
thestandardreferences. Thispartisintendedasawarm-upfortherestofthepaper,and
especially as an introduction to the literature for beginners in the field. The experts are
advisedtoskipto§4(withperhapstheexceptionof§2.3,2.4).
Thesecondpartconsistsof§4-§7,andisasurveyofthecurrentstatusinthetheoryof
linearseriesandgeneralizedthetadivisorsonthesemodulispaces,includingtheconnec-
tion with Quotschemes. Beauville’ssurvey[Be2] has beena greatsourceof inspiration
forresearchersinthisfield(andinfactthemainoneformyself). Anumberofproblems
proposedtherehavebeenhowever(atleastpartially)solvedinrecentyears,andnewcon-
jectureshaveemerged.Anupdateisprovidedin[Be5],wheretheemphasisisspecifically
onthethetamap. Heremygoalistocomplementthisbyemphasizingeffectiveresultson
pluri-thetalinearseries,andproposingsomequestionsinthisdirection.
Concretely, I would like to stress relatively new techniques employed in the analysis
of linear series on modulispaces of vector bundles, namely the use of moduli spaces of
stablemapsforunderstandingQuotschemes,andoftheFourier-Mukaifunctorinthestudy
of coherent sheaves on Jacobians coming from generalized theta divisors. In addition,
I will briefly describe recent important developments, namely the proof of the Strange
DualityconjecturebyBelkaleandMarian-Oprea,andthealgebro-geometricderivationof
theVerlindeformulaviaintersectiontheoryonQuotschemes,duetoMarian-Oprea.
ThereiscertainlyrecentandnotsorecentliteraturethatIhavenotcoveredinthissurvey,
forwhichIapologize.Thisismainlyduetothefactthatingoesindirectionsdifferentfrom
themainthrusthere,butsometimesalsotomyownlackofsufficientfamiliaritywiththe
respectiveresults. Forthose soinclined, muchofthematerialcanalso bephrasedin the
languageofstacks,withlittleornodifferenceintheproofsoftheresultsin§3-7.
Acknowledgements. I would like to warmly thank the organizersof the Cologne Sum-
merSchool,ThomasEcklandStefanKebekus,foranextraordinarilywell-organizedand
successfulschool,whichIconsideramodelforsuchevents. I amalso gratefultoallthe
students who took part for their enthusiasm and their helpfulcommentson the material.
Finally, I thank ArnaudBeauville, Prakash Belkale, Alina Marian and Dragos¸ Oprea for
veryusefulcommentsandcorrections.
2. SEMISTABLEBUNDLES
Most of the foundationalmaterial in this section can be found in detail in the books
ofLePotier[LP2]andSeshadri[Se],especiallytheconstructionofmodulispaces,which
is only alluded to here. I have only included sketches of simpler arguments, and some
exercises, in orderto givethe verybeginneran idea of the subject. However,an explicit
descriptionof two specialclasses ofvectorbundlesthatmightbe of interestto the more
advancedisgivenin§2.3and§2.4.
2.1. Arbitraryvectorbundles. LetX bea smoothprojectivecurveofgenusg overan
algebraicallyclosedfieldk. Iwillidentifyfreelyvectorbundleswithlocallyfreesheaves.
Generalizedthetalinearseriesonmodulispacesofvectorbundlesoncurves 3
Definition 2.1. Let E be a vectorbundle on X, of rank r. The determinant of E is the
linebundledetE := ∧rE. ThedegreeofE isthedegreeeofdetE.1 TheslopeofE is
µ(E)= e ∈Q.
r
TheRiemann-Rochformulaforvectorbundlesoncurvessays:
χ(E)=h0(X,E)−h1(X,E)=e+r(1−g).
Animportantexampleisthefollowing:χ(E)=0 ⇐⇒ µ(E)=g−1.
Exercise2.2. LetE be a vectorbundleofrankr on thecurveX. Then, foreachk, the
degreeofthequotientbundlesofEofrankkisboundedfrombelow.
From the modulipointofview, the initialidea wouldbe to constructan algebraicva-
riety (orscheme)parametrizingthe isomorphismclasses of all vectorbundleswith fixed
invariants,i.e. rankr anddegreee. Notethatfixingtheseinvariantsisthesameasfixing
theHilbertpolynomialofE.
Definition2.3. LetBbeasetofisomorphismclassesofvectorbundles. WesaythatBis
boundedifthereexistsaschemeoffinitetypeS overk andavectorbundleF onS ×X
suchthatalltheelementsofBarerepresentedbysomeF :=F withs∈S.
s |{s}×X
Wefindeasilythattheinitialideaaboveistoonaive.
Lemma2.4. Thesetofisomorphismclassesofvectorbundlesofrankr anddegreeeon
X isnotbounded.
Proof. Assuming thatthe family is bounded,use the notationin Definition2.3. As F is
flat over S, by the semicontinuity theorem we get that there are only a finite number of
possiblehiF fori = 0,1. Fixnowapointx ∈ X anddefineforeachk ∈ Nthevector
s
bundle
E :=O (−kx)⊕O ((k+e)x)⊕O⊕r−2.
k X X X
Theyallclearlyhaverankr anddegreee. Ontheotherhand,whenk → ∞wesee that
h0E andh1E alsogoto∞,whichgivesacontradiction. (cid:3)
k k
There exists however a well-known bounded moduli problem in this context, which
producestheQuotscheme. LetE beavectorbundleofrankranddegreeeonX,andfix
integers0≤k ≤randd. Wewouldliketoparametrizeallthequotients
E −→Q−→0
withQacoherentsheafofrankkanddegreed.2WeconsidertheQuotfunctor
QuotE : Algebraicvarieties/k→Sets
k,d
associatingto each S the set ofcoherentquotientsofE := p∗ E which are flatoverS
S X
andhaverankkanddegreedovereachs ∈ S. Thisisacontravariantfunctorassociating
to T →f S the map taking a quotient E → Q to the quotient E = (f ×id)∗E →
S T S
(f ×id)∗Q.
1ThisisthesameasthefirstChernclassc1(E).
2Recallthattherankofanarbitrarycoherentsheafisitsrankatageneralpoint,whilethedegreeisitsfirst
Chernclass.
4 MihneaPopa
Theorem2.5(Grothendieck,cf. [LP2]§4). ThereexistsaprojectiveschemeQuot (E)
k,d
offinitetypeoverk,whichrepresentsthefunctorQuotE .
k,d
This means the following: there exists a “universal quotient” E → Q on
Quot (E)
k,d
Quot (E)×X,whichinducesforeachvarietyS anisomorphism
k,d
Hom(S,Quot (E))∼=QuotE (S)
k,d k,d
givenby
(S →f Quot (E))→(E →(f ×id)∗Q).
k,d S
The terminology is: the Quot functor (scheme) is a fine moduli functor (space). I will
discussmorethingsaboutQuotschemeslater–fornowlet’sjustnotethefollowingbasic
fact,whichisastandardconsequenceofformalsmoothness.
Proposition2.6([LP2]§8.2). LetE beavectorbundleofrankranddegreee,and
q : [0→G→E →F →0]
apointinQuot (E). Then:
k,d
(1)ThereisanaturalisomorphismTqQuotk,d(E)∼=Hom(G,F)(∼=H0(G∨⊗F)).
(2)IfExt1(G,F)(∼=H1(G∨⊗F))=0,thenQuot (E)issmoothatq.
k,d
(3)Wehave
h0(G∨⊗F)≥dim Quot (E)≥h0(G∨⊗F)−h1(G∨⊗F).
q k,d
Thelastquantityisχ(G∨⊗F)=rd−ke−k(r−k)(g−1)(byRiemann-Roch).
2.2. Semistable vectorbundles. To remedythe problemexplainedin the previoussub-
section,oneintroducesthefollowingnotion.
Definition 2.7. Let E be a vector bundle on X of rank r and degree e. It is called
semistable(respectivelystable)ifforanysubbundle06=F ֒→E,wehaveµ(F)≤µ(E)
(respectivelyµ(F)<µ(E)). Itcanbeeasilycheckedthatinthedefinitionwecanreplace
subbundleswitharbitrarycoherentsubsheaves.
Exercise2.8. LetEandF bevectorbundlessuchthatχ(E⊗F)=0. IfH0(E⊗F)=0
(orequivalently,byRiemann-Roch,H1(E⊗F)=0),thenE andF aresemistable.
Exercise2.9. LetF beastablebundleonthecurveX. Foranyexactsequence
0−→E −→F −→G−→0
wehaveHom(G,E)=0.
Herearesomebasicproperties.
Proposition2.10. If E and F are stable vector bundles and µ(E) = µ(F), then every
non-zerohomomorphismφ : E → F isanisomorphism. InparticularHom(E,E) ∼= k
(i.e.E issimple).
Generalizedthetalinearseriesonmodulispacesofvectorbundlesoncurves 5
Proof. Say G = Im(φ). Then by definition we must have µ(E) ≤ µ(G) ≤ µ(F) and
whereverwe have equality the bundlesthemselvesmustbe equal. Since µ(E) = µ(F),
wehaveequalityeverywhere,whichimplieseasilythatφmustbeanisomorphism. Now
ifφ ∈ Hom(E,E), bytheabovewesee thatk[φ] isanfinite fieldextensionofk. Since
thisisalgebraicallyclosed,wededucethatφ=λ·Id,withλ∈k∗. (cid:3)
Exercise2.11. Fixaslopeµ∈Q,andletSS(µ)bethecategoryofsemistablebundlesof
slopeµ. BasedonProposition2.10,showthatSS(µ)isanabeliancategory.
Definition2.12. LetE ∈SS(µ). AJordan-HölderfiltrationofE isfiltration
0=E ⊂E ⊂...⊂E =E
0 1 p
suchthateachquotientE /E isstableofslopeµ.
i+1 i
Proposition2.13. Jordan-Hölderfiltrationsexist. Anytwohavethesamelengthand,upon
reordering,isomorphicstablefactors.
Proof. (Sketch)Sincetherankdecreases,thereisaG⊂E stableofslopeµ. Thisimplies
thatE/GissemistableofslopeµandwerepeattheprocesswithE/GinsteadofE. The
restisawell-knowngeneralalgebraargument. (cid:3)
Definition2.14. (1)ForanyJordan-HölderfiltrationE ofE,wedefine
•
gr(E):=gr(E )= E /E .
• i+1 i
M
i
ThisiscalledthegradedobjectassociatedtoE (well-definedbytheabove).
(2)AvectorbundleEiscalledpolystableifitisadirectsumofstablebundlesofthesame
slope.(SoforE semistable,gr(E)ispolystable.)
(3)TwobundlesE,F ∈SS(µ)arecalledS-equivalentifgr(E)∼=gr(F).
Sometimeswecanreducethestudyofarbitrarybundlestothatofsemistableonesvia
thefollowing:
Exercise2.15(Harder-Narasimhanfiltration). LetE beavectorbundleonX. Thenthere
existsanincreasingfiltration
0=E ⊂E ⊂...⊂E =E
0 1 p
suchthat
(1) EachquotientE /E issemistable.
i+1 i
(2) Wehaveµ(E /E )>µ(E /E )foralli.
i i−1 i+1 i
Thefiltrationisunique;itiscalledtheHarder-NarasimhanfiltrationofE.
Example2.16. (1)Alllinebundlesarestable. AnyextensionofvectorbundlesinSS(µ)
isalsoinSS(µ).
(2)If(r,e)=1,thenstableisequivalenttosemistable.
(3) If X = P1, by Grothendieck’s theorem we know that every vector bundle splits as
E ∼=O(a1)⊕...⊕O(ar),soitissemistableiffallaiareequal.
6 MihneaPopa
(4)We willencountersomeveryinterestingexamplesbelow. Untilthen, hereisthefirst
type of examplewhich requiresa tiny argument. Say L and L are line bundleson X,
1 2
withdegL =danddegL =d+1. Considerextensionsoftheform
1 2
0−→L −→E −→L −→0.
1 2
TheseareparametrizedbyExt1(L ,L )∼=H1(X,L ⊗L−1). ByRiemann-Rochthisis
2 1 1 2
isomorphictokg−2,soassoonasg ≥ 3wecanchoosetheextensiontobenon-split. For
suchachoiceEisstable:firstnotethatµ(E)=d+1/2.ConsideranylinesubbundleM
ofE. IfdegM ≤deverythingisfine. Ifnot,theinducedmapM →L mustbenon-zero
2
(otherwiseM wouldfactorthroughL ,oftoolowdegree).Thisimmediatelyimpliesthat
1
itmustbeanisomorphism,whichisacontradictionsincetheextensionisnon-split.
For later reference, let me also mention the following important result. An elementary
proofbasedonGieseker’sideascanbefoundin[Laz2]§6.4.
Theorem2.17. Assumethatchar(k)=0.IfEandF aresemistablebundles,thenE⊗F
isalsosemistable.3Inparticular,foranyk,SkE and∧kE arealsosemistable.
2.3. Example: Lazarsfeld’s bundles. Here are the more interesting examples of
semistable bundles promised above. In this subsection, following [Laz1], I describe a
constructionconsideredbyLazarsfeldinthestudyofsyzygiesofcurves.InthenextIwill
describeadifferentconstructionduetoRaynaud.
ConsideralinebundleLonX ofdegreed≥ 2g+1. DenotebyM thekernelofthe
L
evaluationmap:
0−→M −→H0(L)⊗O −e→v L−→0
L X
andletQ =M∨. NotethatrkQ =h0L−1=d−ganddegL=d,soµ(Q )= d .
L L L L d−g
ThemainpropertyofQ isthefollowing:
L
Proposition 2.18. If x ,...,x are the points of a generic hyperplane section of X ⊂
1 d
P(H0(L)),thenQ sitsinanextension:
L
d−g−1
0−→ O (x )−→Q −→O (x +...+x )−→0.
X i L X d−g d
M
i=1
Proof. (1)Lookfirstatageneralsituation: assumethatx ,...,x aredistinctpointson
1 k
X,withD =x +...+x ,suchthat:
1 k
(1) L(−D)isgloballygenerated.
(2) h0L(−D)=h0L−k.
Claim: Inthiscasewehaveanexactsequence
k
0−→M −→M −→ O (−x )−→0.
L(−D) L X i
M
i=1
3ItisalsotruethatifEandF areactuallystable,thenE⊗F ispolystable.
Generalizedthetalinearseriesonmodulispacesofvectorbundlesoncurves 7
Proof. By Riemann-Roch and (1) we have h1L(−D) = h1L. This implies that after
passingtocohomology,theexactsequence
0−→L(−D)−→L−→L −→0
D
staysexact. NotealsothattheevaluationmapforL canbewrittenas
D
k
H0L ⊗O →L = (O →O ) .
D X D X xi
(cid:0) (cid:1) (cid:0)Mi=1 (cid:1)
Using all of these facts, (2) and the Snake Lemma, we obtain a diagram whose top row
givespreciselyourClaim.
0 0 0
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
0 // ML(−D) //ML //⊕ki=1OX(−xi) //0
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
0 // H0L(−D)⊗OX //H0L⊗OX //H0LD //0
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
0 //L(−D) // L //LD //0
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
0 0 0
(cid:3)
(2) Note now that if in the situation above we also have h0L(−D) = 2, then the exact
sequencetakestheform
k
0−→O (x +...+x )⊗L−1 −→M −→ O (−x )−→0.
X 1 k L X i
M
i=1
Thisisclear,sinceinthiscasebydefinitionrkM =1anddetM =L−1(D).
L(−D) L(−D)
(3)SaynowthatX isembeddedinPnbyalinebundleL,andΛ:=Span x ,...,x ⊂
1 k
Pn. Thenwehavethefollowinggeneral (cid:0) (cid:1)
Exercise2.19. (a)h0L(−D)=h0L−k ⇐⇒ dimΛ=k−1.
(b)L(−D)isgloballygenerated ⇐⇒ Λ∩X ={x ,...,x }andΛdoesnotcontainthe
1 k
tangentlinetoX atx foralli.
i
(4)NowconsiderthesituationintheProposition,whenX ⊂P(H0L)andx ,...,x are
1 d
thepointsofageneralhyperplanesectionofX. Thisimpliesthatthepointsareinlinear
generalposition,whichimmediatelygivesthattheconditionsinExercise2.19aresatisfied.
Bythepreviouspoints,wearedone. (cid:3)
8 MihneaPopa
Proposition2.18impliesthestabilityofQ . TheproofbelowisduetoEin-Lazarsfeld
L
[EL1].
Proposition2.20. UndertheassumptionsaboveQ isastablebundle.
L
Proof. Let’s see that the dual M is stable. One can actually prove a bit more: M is
L L
cohomologicallystable,i.e. foranylinebundleAofdegreeaandanyt<rkM =d−g:
L
t
td
(1) H0( M ⊗A−1)=0if a≥t·µ(M )=− .
L L
d−g
^
ThisimpliesthestabilityofM : indeed,ifF ֒→M isasubbundleofdegreeaandrank
L L
t, then we have an inclusion A := tF ֒→ tM , which implies that H0( tM ⊗
L L
A−1) 6= 0. By cohomologicalstabiVlity we muVst have µ(F) = a < µ(M ),Vso M is
t L L
stable.
To prove (1), take exterior powers in the dual of the sequence in Proposition 2.18 to
obtain
t−1 d−g−1 t t d−g−1
0→O (−x −...−x )⊗ ( O (−x ))→ M → ( O (−x ))→0.
X d−g d X i L X i
^ M ^ ^ M
i=1 i=1
Inotherwordswehaveanexactsequence
t
0→ O (−x −...−x −x ...−x ))→ M
X i1 it−1 d−g d L
M ^
1≤i1<...<it−1≤d−g−1
→ O (−x −...−x ))→0.
X j1 jt
M
1≤j1<...<jt≤d−g−1
We tensor this sequence by A−1. It can be checked easily that on both extremes H0 is
zero,asthepointsx aregeneral.Thisimplieswhatwewant. (cid:3)
i
Corollary2.21. Ifchar(k)=0,thenforallpthebundle pQ issemistable.
L
V
2.4. Example: Raynaud’sbundles. (cf. [Ra])LetX ֒→ J(X)be an Abel-Jacobiem-
bedding. Denote A = J(X). Then A has a principalpolarizationΘ, which inducesan
isomorphism A ∼= A = Pic0(A). Denote by P a Poincaré bundle on A×A. For any
m≥1,weconsiderwhatiscalledtheFourier-MukaitransformofOb(−mΘ),namely:
b A b
\
F :=OAb(−mΘ):=RgpA∗(p∗AbO(−mΘ)⊗P).
Bybasechange,thisisavectorbundlewithfiberoverx∈Aisomorphicto
Hg(A,OAb(−mΘ)⊗P|{x}×Ab)∼=H0(J(X),OJ(X)(mΘ)⊗Px)∨,
whereP istheblinebundleinPic0(J(X)))correspondingtox ∈ J(X). (Notethatasx
x
varieswith J(X), P varieswithPic0(J(X)).) HenceF isa vectorbundleofrankmg.
x
DefinethevectorbundleE :=F onX.
|X
Theclaimisthatthisisasemistablebundle. Indeed,considerthemultiplicationbym
mapφ :A→A. ByaresultofMukai,[Mu2]Proposition3.11,wehavethat
m
φ∗mF ∼=H0OAb(mΘ)⊗OAb(mΘ)∼= OAb(mΘ).
Mmg
Generalizedthetalinearseriesonmodulispacesofvectorbundlesoncurves 9
Weconsidertheétalebasechangeψ : Y →X,whereY = φ−1(X). Thedecomposition
m
ofF viapull-backbyφ impliesthatψ∗E issemistableonY. Applyingthewell-known
m
Lemmabelow,wededucethatEissemistable.
Lemma2.22. (cf. [Laz2]Lemma6.4.12)Letf : Y →X beafinitemorphismofsmooth
projectivecurves,andE avectorbundleonX.ThenEissemistableifandonlyiff∗Eis
semistable.
Let’salsocomputetheslopeofE.Notethatdegψ∗E =degφ ·degE =m2g·degE.
m
Thisgives
mg·m·(θ·[Y]) θ·[Y]
degE = = .
m2g mg−1
ButsinceY =φ−1(X),φ∗ θ ≡m2·θandθ·[X]=g,wehavethatm2·(θ·[Y])=g·m2g.
m m
Puttingeverythingtogether,wefinallyobtain
g
degE =g·mg−1, i.e.µ(E)= .
m
Remark2.23. Schneider[Sch2]usesaslightlymorerefinedstudyofRaynaud’sbundles
basedontheta-groupactionsinordertoshowthattheyareactuallystable.
Remark2.24. AgeneralizationofRaynaud’sexamples,bymeansofsemi-homogeneous
vectorbundlesontheJacobianofX,ispresentedinRemark7.18(2).
Remark2.25. Iwillonlymentioninpassingonemoreinterestingidea,whichhastodo
withmoduliofcurves:certaintypesofstablebundlescanexistonlyonspecialcurves,just
like in the usual Brill-Noether theory for line bundles. One can look for vector bundles
with“many"sections,i.e. wonderwhetherforagivenkthereexistsemistablebundlesE
onX ofrankranddegreed(orfixeddeterminantL),whichhavekindependentsections.
For instance, if g(X) = 10, one can see that the condition that there exist a semistable
bundleofrank2anddeterminantω withatleast7sectionsiscodimension1inM ,i.e.
X 10
suchbundlesexistonlyoncurvesfillingupadivisorinM . Theclosureofthislocusin
10
M isaveryinterestingdivisor,thefirstoneshowntobeofslopesmallerthanexpected
10
(cf. [FP]). As examples of stable bundles though, the vector bundles here are not quite
new:theyareroughlyspeakingofthesamekindastheLazarsfeldexamplesabove.
2.5. Themodulispace. Backtotheboundednessproblem:wewanttoseethatsemistable
bundlesdothejob.Firstatechnicalpoint,extendingawell-knownfactforlinebundles.
Exercise2.26. LetEbeasemistablebundleonX.
(a)Ifµ(E)>2g−2,thenh1E =0.
(b)Ifµ(E)>2g−1,thenE isgloballygenerated.
Proposition2.27. ThesetS(r,d)ofisomorphismclassesofsemistablebundlesofrankr
anddegreedisbounded.
Proof. FixO (1)apolarizationonthecurve.ByExercise2.26,thereexistsafixedm≫
X
0suchthatforallF inS(r,d)wehaveh1F(m)=0andF(m)isgloballygenerated.Let
q :=h0F(m)=χ(F(m)),whichisconstantbyRiemann-Roch.Theglobalgenerationof
F(m)meansthatwehaveaquotient
O⊕q(−m)−β→F −→0.
X
10 MihneaPopa
TheseallbelongtotheQuotschemeQuot (O⊕q(−m)),whichisaboundedfamily. (cid:3)
r,d X
Thequotientβ canberealizedinmanyways: fixa vectorspaceV ∼= kq, andchoose
anisomorphismV ∼= H0F(m). OnQuotr,d(V ⊗OX(−m))∼=Quotr,d(OX⊕q(−m))we
haveanaturalGL(V)-action,namelyeachg ∈GL(V)inducesadiagram
V g⊗⊗OidX(cid:15)(cid:15) (−ssmss)sssssss//99 Q
V ⊗O (−m)
X
Thescalarmatricesacttriviallyonthesequotients,soinfactwehaveaPGL(V)-action.
Proposition2.28. LetΩ⊂Quot (V ⊗O (−m))bethesetofquotientsQsuchthatQ
r,d X
issemistableandV ∼= H0Q(m). ThenΩisinvariantunderthePGL(V)-action,andwe
haveabijection
Ω/PGL(V)−→S(r,d).
Proof. The set Ω is clearly invariant, and the points in the same orbit give isomorphic
quotientbundles, sowe havea naturalmapΩ/PGL(V) −→ S(r,d). Since m ≫ 0, by
globalgenerationthemapissurjective. Supposenowthatwehavetwodifferentquotients
inducinganisomorphismφ:
V ⊗O (−m) //Q
X
φ ∼=
(cid:15)(cid:15)
V ⊗O (−m) //Q
X
This inducesan isomorphismH0Q(m) H−0φ→(m) H0Q(m), which correspondsto an ele-
mentg ∈GL(V). Thisisuniquelydetermineduptoscalars,sowecaninfactconsiderit
inPGL(V). (cid:3)
Although this was not one of the topics of the lectures, the main point of Geometric
InvariantTheory(GIT)– appropriatelyforthisvolume,oneofD. Mumford’smostcele-
brated achievements– is in this contextessentially to show that in fact this quotienthas
the structure of a projective algebraic variety.4 The GIT machineryconstructs the space
U (r,d) –analgebraicvarietyreplacementofS –themodulispaceofS-equivalence
X r,d
classes of semistable vector bundles of rank r and degree d on X. For the well-known
construction,thatcanbeadaptedtoarbitrarycharacteristicwithoutdifficulty,cf. [LP2]§7
or[Se]Ch.I.WedenotebyUs(r,d)theopensubsetcorrespondingtoisomorphismclasses
X
ofstablebundles.
We consideralso a variantof U (r,d) when the determinantof the vectorbundlesis
X
fixed. Moreprecisely,foranyL ∈ Picd(X), wedenotebySU (r,L)themodulispace
X
of(S-equivalenceclassesof)semistablebundlesofrankrandfixeddeterminantL. These
arethefibersofthenaturaldeterminantmap
det:U (r,d)−→Picd(X).
X
4ThisisliterallytrueifweconsiderS-equivalenceclassesofsemistablebundles(cf.Definition2.14)instead
ofisomorphismclasses.