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Vector bundles and theta functions on curves of genus 2 and 3

Arnaud Beauville
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Abstract

Let C be a curve of genus g, and let SU(r) be the moduli space of vector bundles of rank r on C, with trivial determinant. A general E in SU(r) defines a theta divisor in the linear system |r Theta|, where Theta is the canonical theta divisor in Pic^{g-1}(C). This defines a rational map SU(r) - - > |r Theta|, which is the map associated to the determinant bundle on SU(r) (the positive generator of Pic(SU(r)). In this paper we prove that in genus 2 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 E in SU(3) admits a theta divisor.

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Arnaud Beauville. 2004-06-02. Vector bundles and theta functions on curves of genus 2 and 3. https://arxiv.org/abs/math/0406030

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