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 ΘE in the linear system |rΘ|, where Θ is the canonical theta divisor in Picg-1 (C). This defines a rational map Θ: SUC(r) → |rΘ|, 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.


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pp. 607-618
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