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

Let SU_C(r) be the moduli space of vector bundles of rank r and trivial determinant on a curve C. A general E in SU_C(r) defines a divisor Θ_E in the linear system |rΘ|, where Θ is the canonical theta divisor in Pic^g-1 (C). This defines a rational map Θ: SU_C(r) → |rΘ|, which turns out to be the map associated to the determinant bundle on SU_C(r) (the positive generator of Pic (SU_C(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 SU_C(3) admits a theta divisor.