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 nonisomorphic 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–Macrì divisor associated to the wall is a boundary nef divisor on the moduli space of sheaves M_H(v). We recover previous results for ℙ² and K3 surfaces, and illustrate applications to higher Picard rank surfaces with an example on ℙ¹ × ℙ¹.