Seshadri constants on symmetric products of curves

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Let Xg=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 Xg to a problem involving the Seshadri constant of a point on Xg₋₁. Using this we recover a result of Ciliberto-Kouvidakis that reduces finding the ample cone of Xg to the Nagata conjecture when g≥ 9. We also give new bounds on the the ample cone of Xg when g=5.