Effective cone of the blowup of the symmetric product of a curve

Abstract

Let C C be a smooth curve of genus g ≥ 1 g \geq 1 and let C ( 2 ) C^{(2)} be its second symmetric product. In this note we prove that if C C is very general, then the blowup of C ( 2 ) C^{(2)} at a very general point has nonpolyhedral pseudo-effective cone. The strategy is to consider first the case of hyperelliptic curves and then to show that having polyhedral pseudo-effective cone is a closed property for families of surfaces.