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Fibrés de rang 2 sur une courbe, fibré déterminant et fonctions thêta. (Rank 2 fiber bundles on a curve, determinant bundle and theta functions). (French) Zbl 0691.14016

Summary: Let \({\mathcal N}\) be the moduli space of semi-stable rank 2 vector bundles with trivial determinant on a curve C. We prove that the Picard group of \({\mathcal N}\) is generated by the determinant bundle \({\mathcal L}\). We identify the space of global sections of \({\mathcal L}\) with the space of 2nd order theta functions on the Jacobian J of C, so that \({\mathcal L}\) defines a morphism of \({\mathcal N}\) into the linear system \(| 2\theta |\) on J. We give a geometric interpretation of this morphism, and prove that it is of degree 2 if C is hyperelliptic and of degree 1 otherwise.

MSC:

14H10 Families, moduli of curves (algebraic)
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14K25 Theta functions and abelian varieties
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