The Nef Cone of the Moduli Space of Sheaves and Strong Bogomolov Inequalities

Let (X, H) be a polarized, smooth, complex projective surface, and let v be a Chern character on X with positive rank and sufficiently large discriminant. In this paper, we compute the Gieseker wall for v in a slice of the stability manifold of X. We construct explicit curves parameterizing non-isomorphic Gieseker stable sheaves of character v that become S-equivalent along the wall. As a corollary, we conclude that if there are no strictly semistable sheaves of character v, the Bayer-Macr̀ı divisor associated to the wall is a boundary nef divisor on the moduli space of sheaves MH(v). We recover previous results for P and K3 surfaces, and illustrate applications to higher Picard rank surfaces with an example on P × P.