The nef cone of the moduli space of sheaves and strong Bogomolov inequalities

TL;DR

This paper analyzes the nef cone of the moduli space of sheaves on complex surfaces, computes stability walls, and establishes strong Bogomolov inequalities, extending known results to higher Picard rank surfaces.

Contribution

It explicitly computes the Gieseker wall for certain Chern characters and links stability conditions to nef divisors on moduli spaces, generalizing previous results.

Findings

Computed the Gieseker wall for large discriminant sheaves.

Constructed explicit curves of Gieseker stable sheaves becoming S-equivalent.

Identified boundary nef divisors related to stability walls.

Abstract

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 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-Macri divisor associated to the wall is a boundary nef divisor on the moduli space of sheaves M_H(v). We recover previous results for the projective plane and K3 surfaces, and illustrate applications to higher Picard rank surfaces with an example on a quadric surface.