New constructions of nef classes on self-products of curves

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.