Vector bundles and theta functions on curves of genus and

Arnaud Beauville

Description

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 ? ?

Sujets

• Beauville

Informations

• Publié par: pefav
• Nombre de lectures: 25
• Langue: English

Extrait

Vectorbundlesandthetafunctionsoncurvesofgenus2and3 ArnaudB EAUVILLE Abstract Let SU C (r)bethemodulispaceofvectorbundlesofrankrand trivialdeterminantonacurveC.AgeneralEin SU C (r)definesadivisor Θ E inthelinearsystem | rΘ | ,whereΘisthecanonicalthetadivisorin Pic g−1 (C).Thisdefinesarationalmapθ: SU C (r) 99K | rΘ | ,whichturns outtobethemapassociatedtothedeterminantbundleon SU C (r)(the positivegeneratorofPic( SU C (r)).Ingenus2weprovethatthismap isgenericallyfiniteanddominant.Thesamemethod,togetherwithsome classicalworkofMorin,showsthatinrank3andgenus3thethetamapis afinitemorphism–inotherwords,everyvectorbundlein SU C (3)admits athetadivisor. Introduction LetCbeasmoothprojectivecomplexcurve,ofgenusg ≥ 2.Themodulispace SU C (r)ofsemi-stablevectorbundlesofrankronC,withtrivialdeterminant,is anormalprojectivevariety,whichcanbeconsideredasanon-abeliananalogueof theJacobianvarietyJC.ItisactuallyrelatedtoJCbythefollowingconstruction, whichgoesback(atleast)to[N-R].LetJ g − 1 bethetranslateofJCparameterizing linebundlesofdegreeg − 1onC,andΘ ⊂ J g − 1 thecanonicalthetadivisor.For E ∈SU C (r),considerthelocus Θ E := { L ∈ J g − 1 | H 0 (C,E ⊗ L) 6 =0 } . TheneitherΘ E =J g − 1 ,orΘ E isinanaturalwayadivisorinJ g − 1 ,belongingto thelinearsystem | rΘ | .Inthiswaywegetarationalmap θ: SU C (r)99K | rΘ | whichisthemostobviousrationalmapof SU C (r)inaprojectivespace:itcan beidentifiedtothemapϕ L : SU C (r)99K P (H 0 ( SU C (r), L ) ∗ )givenbytheglobal sectionsofthedeterminantbundle L ,thepositivegeneratorofthePicardgroupof SU C (r)[B-N-R]. Forr=2themapθisanembeddingifCisnothyperelliptic[vG-I].We considerinthispaperthehigherrankcase,whereverylittleisknown.Thefirstpart isdevotedtothecaseg=2.Thereacuriousnumericalcoincidenceoccurs,namely dim SU C (r)=dim | rΘ | =r 2 − 1. 1