  We study the nef cone of self-products of a curve. When the curve is very
general of genus $g>2$, we construct a nontrivial class of self-intersection 0
on the boundary of the nef cone. Up to symmetry, this is the only known
nontrivial boundary example that exists for all $g > 2$. When the curve is
general, we identify nef classes that improve on known examples for arbitrary
curves. We also consider self-products of more than two copies of the curve.
