The Brill-Noether curve of a stable vector bundle on a genus two curve

Let SU_X(r,0) be the moduli space of rank r semistable vector bundles with trivial determinant over a smooth, integral complex curve of genus g > 1. The theta map t_r: SU_X(r,0) \to P^N is the rational map associated to the ample generator of Pic SU_X(r,0). T_r is known to have indeterminacy for r >> 0. For genus g = 2 it turns out that N is the dimension of SU_X(r,0). Let g = 2, in the paper a geometric description of t_r is given. This description implies the generic finiteness of t_r: a result obtained independently by A. Beauville for g = 2, 3 in the paper . Let E be a general stable vector bundle on X defining a point x of SU_X(r,0) and let D_E its associated divisor in Pic^1 (X). The fibre of t_r at x is described as the set of the irreducible components of a suitable Brill-Nether locus of the curve D_E. This implies the generic finiteness.