Seshadri constants on symmetric products of curves

Abstract

Let $X_g=C^{(2)}_g$ be the second symmetric product of a very general curve of genus $g$. We reduce the problem of describing the ample cone on $X_g$ to a problem involving the Seshadri constant of a point on $X_{g-1}$. Using this we recover a result of Ciliberto-Kouvidakis that reduces finding the ample cone of $X_g$ to the Nagata conjecture when $g\ge 9$. We also give new bounds on the the ample cone of $X_g$ when $g=5$.