Some surfaces with non-polyhedral nef cones

Abstract

Over algebraically closed fields of positive characteristic, C×C contains the family of curves cut out as graphs of iterates of the Frobenius morphism. Their self-intersection tends to negative infinity; hence C × C fails the bounded negativity conjecture in positive characteristic. This work arises as part of the broader question of whether C×C where C has genus ≥ 2 satisfies the bounded negativity conjecture in characteristic 0. When a surfaceX has a polyhedral nef cone, it automatically satisfies bounded negativity since all negative curves are extremal and polyhedral cones have finitely many extremal rays. We establish that this naive approach to bounded negativity fails for C×C. The original question remains open.