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Vector bundles and theta functions on curves of genus and

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12 Pages
English

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Vector bundles and theta functions on curves of genus 2 and 3 Arnaud BEAUVILLE Abstract Let SUC(r) be the moduli space of vector bundles of rank r and trivial determinant on a curve C . A general E in SUC(r) defines a divisor ?E in the linear system |r?| , where ? is the canonical theta divisor in Picg?1(C) . This defines a rational map ? : SUC(r) 99K |r?| , which turns out to be the map associated to the determinant bundle on SUC(r) (the positive generator of Pic(SUC(r)) . In genus 2 we prove that this map is generically finite and dominant. The same method, together with some classical work of Morin, shows that in rank 3 and genus 3 the theta map is a finite morphism – in other words, every vector bundle in SUC(3) admits a theta divisor. Introduction Let C be a smooth projective complex curve, of genus g ≥ 2 . The moduli space SUC(r) of semi-stable vector bundles of rank r on C , with trivial determinant, is a normal projective variety, which can be considered as a non-abelian analogue of the Jacobian variety JC . It is actually related to JC by the following construction, which goes back (at least) to [N-R].

  • let ?

  • symplectic bundles

  • semi-stable vector

  • classical projective

  • canonical theta

  • theta-characteristic ? ?


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Vectorbundlesandthetafunctionsoncurvesofgenus2and3ArnaudBEAUVILLEAbstractLetSUC(r)bethemodulispaceofvectorbundlesofrankrandtrivialdeterminantonacurveC.AgeneralEinSUC(r)definesadivisorΘEinthelinearsystem|rΘ|,whereΘisthecanonicalthetadivisorinPicg1(C).Thisdefinesarationalmapθ:SUC(r)99K|rΘ|,whichturnsouttobethemapassociatedtothedeterminantbundleonSUC(r)(thepositivegeneratorofPic(SUC(r)).Ingenus2weprovethatthismapisgenericallyfiniteanddominant.Thesamemethod,togetherwithsomeclassicalworkofMorin,showsthatinrank3andgenus3thethetamapisafinitemorphism–inotherwords,everyvectorbundleinSUC(3)admitsathetadivisor.IntroductionLetCbeasmoothprojectivecomplexcurve,ofgenusg2.ThemodulispaceSUC(r)ofsemi-stablevectorbundlesofrankronC,withtrivialdeterminant,isanormalprojectivevariety,whichcanbeconsideredasanon-abeliananalogueoftheJacobianvarietyJC.ItisactuallyrelatedtoJCbythefollowingconstruction,whichgoesback(atleast)to[N-R].LetJg1bethetranslateofJCparameterizinglinebundlesofdegreeg1onC,andΘJg1thecanonicalthetadivisor.ForE∈SUC(r),considerthelocusΘE:={LJg1|H0(C,EL)6=0}.TheneitherΘE=Jg1,orΘEisinanaturalwayadivisorinJg1,belongingtothelinearsystem|rΘ|.Inthiswaywegetarationalmapθ:SUC(r)99K|rΘ|whichisthemostobviousrationalmapofSUC(r)inaprojectivespace:itcanbeidentifiedtothemapϕL:SUC(r)99KP(H0(SUC(r),L))givenbytheglobalsectionsofthedeterminantbundleL,thepositivegeneratorofthePicardgroupofSUC(r)[B-N-R].Forr=2themapθisanembeddingifCisnothyperelliptic[vG-I].Weconsiderinthispaperthehigherrankcase,whereverylittleisknown.Thefirstpartisdevotedtothecaseg=2.Thereacuriousnumericalcoincidenceoccurs,namelydimSUC(r)=dim|rΘ|=r21.1