On Pryms rank bundles and nonabelian theta functions

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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


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Nombre de lectures 18
Langue English
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1
On
Pryms,
rank
Introduction
2
bundles and functions
Christian Pauly
september 1995
nonabelian theta
LetM0(resp.Mp) denote the moduli space parametrizing semistable rank 2 bundles with determinant equal toOC(resp.OC(p)) over a smooth, projective curveCof genus g2;pis a xed point ofC. The Picard group of both moduli spaces is isomorphic to ZZ and we denote byL(resp.Lp Then the Verlinde formula gives) their ample generators. the dimension of the vector spacesH0(M0,Lk) andH0(Mp,Lpk) which consist of what is called generalized or nonabelian theta functions of levelk. Several authors have studied the geometry of the moduli spaceM0in connection with the JacobianJand the Prym varietyPxof 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 toM0and 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 withM0, the moduli spaceMpalso contains the Prym varietiesPxand ˆ ˆ (a blown-up of) the JacobianJ observe that the varieties. WeJandPxintersect and that two orthogonal PrymsPxandPy, 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 m2:S2H0(Mp,Lp)→H0(Mp,Lp2)
is surjective
As was shown in [O-P], surjectivity ofm2implies that the natural homomorphism MH0(Px,O(3x))→H0(Mp,Lp2) xJ[2]
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