  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.
