A structure theorem for SU_C(2) and the moduli of pointed rational curves

Let SU_C(2) be the moduli space of rank 2 semistable vector bundles with trivial determinant on a smooth complex algebraic curve C of genus g > 1, we assume C non hyperelliptic if g > 2. In this paper we construct large families of pointed rational normal curves over certain linear sections of SU_C(2). This allows us to give an interpretation of these subvarieties of SU_C(2) in terms of the moduli space of curves M_0,2g. In fact there exists a natural linear map SU_C(2) ----> P^g with modular meaning, whose fibers are birational to M_0,2g, the moduli space of 2g-pointed genus zero curves.