On symmetric products of curves

Let C be a smooth complex projective curve of genus g and let X be its second symmetric product. This paper concerns the study of some attempts at extending to X the notion of gonality. In particular, we prove that the degree of irrationality of X is at least g-1 when C is a generic curve, and that the minimum gonality of curves through the generic point of X 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 \mathbb{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 X when C is a generic curve of genus 5<g<9.