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SU Verlinde spaces as theta spaces on Pryms

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Niveau: Supérieur, Doctorat, Bac+8
SU(2)-Verlinde spaces as theta spaces on Pryms W.M. Oxbury and C. Pauly 1 Introduction By an (SU(r)-)Verlinde space one means a complex vector space H0(M,Lk) of sections of a line bundle Lk on a moduli space M = M(r, d) of semistable vector bundles of rank r and fixed determinant, degree d ? Z/r, over a smooth complex projective curve C. One knows that the projective variety M has infinite cyclic Picard group [DN], and L denotes the ample generator. These spaces,whose dimensions are given by the celebrated Verlinde formulae, are of considerable interest and have been much studied in recent years. The purpose of the present article is to explain the following coincidence in the Verlinde formulae for the anticanonical line bundles of the varieties M0 = M(2, 0) and M1 = M(2, 1). We should first remark that there are already a number of well-known ‘coincidences' linking the rank 2 Verlinde spaces for low values of k to spaces of theta functions on the Jacobian of the curve. For example, the simplest Verlinde formula h0(M0,L) = 2g (where g is the genus of C) was first shown directly [B1] by not- ing that H0(M0,L) is naturally isomorphic to the 2-theta space V = H0(Jg?1(C), 2?).

  • vanishing theta-nulls

  • p?x ?

  • unram- ified double

  • theta functions

  • then ?

  • schottky-jung geometry

  • has infinite cyclic

  • line bundles


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SU(2)-VerlindespacesasthetaspacesonsmyrPW.M.OxburyandC.Pauly1IntroductionByan(SU(r)-)VerlindespaceonemeansacomplexvectorspaceH0(M,Lk)ofsectionsofalinebundleLkonamodulispaceM=M(r,d)ofsemistablevectorbundlesofrankrandfixeddeterminant,degreedZ/r,overasmoothcomplexprojectivecurveC.OneknowsthattheprojectivevarietyMhasinfinitecyclicPicardgroup[DN],andLdenotestheamplegenerator.Thesespaces,whosedimensionsaregivenbythecelebratedVerlindeformulae,areofconsiderableinterestandhavebeenmuchstudiedinrecentyears.ThepurposeofthepresentarticleistoexplainthefollowingcoincidenceintheVerlindeformulaefortheanticanonicallinebundlesofthevarietiesM0=M(2,0)andM1=M(2,1).Weshouldfirstremarkthattherearealreadyanumberofwell-known‘coincidences’linkingtherank2VerlindespacesforlowvaluesofktospacesofthetafunctionsontheJacobianofthecurve.Forexample,thesimplestVerlindeformulah0(M0,L)=2g(wheregisthegenusofC)wasfirstshowndirectly[B1]bynot-ingthatH0(M0,L)isnaturallyisomorphictothe2-thetaspaceV=H0(Jg1(C),2Θ).Thenextcaseisthepairofformulaeh0(M0,L2)=2g1(2g+1),h0(M1,L)=2g1(2g1);thatis,thesegivethenumbersofevenandoddtheta-characteristicsonthecurve.ThisalsowasexplainedbyBeauville[B2]:forgenericCtherespectiveVerlindespacesarecanonicallyisomorphictothe4-thetaspaces1
S2V=H+0(J,4Θ)andV2V=H0(J,4Θ),thesubscript±denotingeven/oddthetafunctions.OnepointofinterestoftheseresultsistheirconnectionwiththeclassicalSchottky-JunggeometryoftheJacobiananditsassociatedPrymvarietiesofunramifieddoublecoversofC.Indeed,theKummersofalltheseabelianvarieties,2-thetaembeddedinP(V),lienaturallyinsidetheimageofM0,whichlives,viatheBeauvilleisomorphism,inthesameprojectivespacebymeansofthelinearsystem|L|.Thisobservationhasbeenused[vGP1]togiveavectorbundletheoreticproofoftheSchottky-Jungrelationsbetweenthetheta-nullsoftheJacobianandthoseofitsPryms.Itisthereforeinterestingtoexplorefurtherthissortofrelationshipbe-tweenthe‘Verlinde’geometryofthecurve,anditsclassicalthetageometry.In[vGP1]thespaceH0(M0,L4)hasbeenstudiedbyrestrictingsectionstoalltheembeddedKummers;thisgivesaninjectionofH0(M0,L4)intothedirectsumofthespacesH+0(Px,x)asxrangesthroughthegroupJ[2]of2-torsionpointsintheJacobian.Thenotationhereis:(Px,Ξx)denotesthecanonicallyprincipallypolarisedPrymvarietyassociatedtothe2-torsionpointxwhenthisisnonzero,andequals(J(C),Θ)whenx=0.Inthepresentworkweattempttoimproveonthis.Oneobservesthefollowing:first,theVerlindeformulaeforh0(M0,L4)andh0(M1,L2),respec-tively,arethefollowingnumbers,differingbyasignchange:3g122g1±22g1+3g13g±12g³3g1±1´=+(21).22Ontheotherhand,foranyprincipallypolarisedabelianvariety(P,Ξ)ofdimensionnonehash0±(P,3Ξ)=(3n±1)/2.Finally,M0andM1areFanovarieties,withanticanonicalsheafgiven(by[DN])byKM01=L4andKM11=2.LTakentogether,theseobservationsmotivatethefollowingresult:1.1Theorem.ForanycurveCtherearenaturalhomomorphisms:LS0:H0(M0,K1)−→xJ[2]H+0(Px,x),LS1:H0(M1,K1)−→xJ[2]H0(Px,x).Moreover,S0isanisomorphismforallCwithoutvanishingtheta-nulls.2