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

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