On Pryms rank bundles and nonabelian theta functions

Christian Pauly

Description

Niveau: Supérieur, Doctorat, Bac+8
On Pryms, rank 2 bundles and nonabelian theta functions Christian Pauly september 1995 1 Introduction Let M0 (resp. Mp) denote the moduli space parametrizing semistable rank 2 bundles with determinant equal to OC (resp. OC(p)) over a smooth, projective curve C of genus g ≥ 2; p is a fixed point of C. The Picard group of both moduli spaces is isomorphic to ZZ and we denote by L (resp. Lp) their ample generators. Then the Verlinde formula gives the dimension of the vector spaces H0(M0,Lk) and H0(Mp,Lkp) which consist of what is called generalized or nonabelian theta functions of level k. Several authors have studied the geometry of the moduli space M0 in connection with the Jacobian J and the Prym variety Px of an unramified double cover of the curve C associated to a nonzero 2-torsion point x. The Kummers of all these abelian varieties can be mapped naturally to M0 and the intersection points of two distinct Kummers give the Schottky-Jung and Donagi relations between their theta-nulls [vG-P1]. As a consequence of these identities, van Geemen and Previato [vG-P m4 : S4H0(M0,L) ?? H0(M0,L4) is surjective. In analogy with M0, the moduli space Mp also contains the Prym varieties Px and (a blown-up of) the Jacobian J .

• degenerate skew-symmetric


• m0 ?


• p?x ??


• half-space such


• theta functions


• px


• map ?x


• bilinear pairing

Sujets

• Theta function

Informations

• Publié par: profil-zyak-2012
• Nombre de lectures: 18
• Langue: English

Extrait

1

On

Pryms,

rank

Introduction

2

bundles and functions

Christian Pauly

september 1995

nonabelian theta

LetM 0 (resp.M p ) denote the moduli space parametrizing semistable rank 2 bundles with determinant equal toO C (resp.O C (p)) over a smooth, projective curveCof genus g2;pis a xed point ofC. The Picard group of both moduli spaces is isomorphic to ZZ and we denote byL(resp.L p  Then the Verlinde formula gives) their ample generators. the dimension of the vector spacesH 0 (M 0 ,L k ) andH 0 (M p ,L pk ) which consist of what is called generalized or nonabelian theta functions of levelk. Several authors have studied the geometry of the moduli spaceM 0 in connection with the JacobianJand the Prym varietyP x of an unramied double cover of the curveC associated to a nonzero 2-torsion pointx. The Kummers of all these abelian varieties can be mapped naturally toM 0 and the intersection points of two distinct Kummers give the Schottky-Jung and Donagi relations between their theta-nulls [vG-P1]. As a consequence of these identities, van Geemen and Previato [vG-P m 4 :S 4 H 0 (M 0 ,L)→H 0 (M 0 ,L 4 )

is surjective. In analogy withM 0 , the moduli spaceM p also contains the Prym varietiesP x and ˆ ˆ (a blown-up of) the JacobianJ observe that the varieties. WeJandP x intersect and that two orthogonal PrymsP x andP y , although they don’t intersect, verify a geometric property, which lead to new relations among theta-constants (see section 4). Finally, we can adapt the method of [vG-P1,2] to prove the main theorem:

Theorem 1.1For a generic curve, the multiplication map m 2 :S 2 H 0 (M p ,L p )→H 0 (M p ,L p 2 )

is surjective

As was shown in [O-P], surjectivity ofm 2 implies that the natural homomorphism M H  0 (P x ,O(3 x )) ∨ →H 0 (M p ,L p 2 ) x∈J[2]

1