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 $\mathbb{P}^n$ 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 $C^{(2)}$ when $C$ is a generic curve of genus ${6\leq g\leq 8}$.