On symmetric products of curves

Let C be a smooth complex projective curve of genus g and let C (2) be its second symmetric product. This paper concerns the study of some attempts at extending to C (2) the notion of gonality. In particular, we prove that the degree of irrationality of C (2) is at least g - 1 when C is generic and that the minimum gonality of curves through the generic point of C (2) equals the gonality of C. In order to produce the main results we deal with correspondences on the k-fold symmetric product of C, with some interesting linear subspaces of ℙ n enjoying a condition of Cayley-Bacharach type, and with monodromy of rational maps. As an application, we also give new bounds on the ample cone of C (2) when C is a generic curve of genus 6 ≤ g ≤ 8