Projectivity and Birational Geometry of Bridgeland moduli spaces

TL;DR

This paper constructs and analyzes nef divisors on Bridgeland moduli spaces, establishing their ampleness and linking wall-crossing phenomena with the minimal model program, with applications to classical K3 surface moduli spaces.

Contribution

It introduces a natural family of nef divisors on Bridgeland moduli spaces, proving ampleness in generic cases and connecting wall-crossing to the minimal model program.

Findings

Constructs nef divisor classes varying with stability conditions.

Proves ampleness of the divisor class for generic stability on K3 surfaces.

Determines nef cones and verifies conjectures for specific K3 moduli spaces.

Abstract

We construct a family of nef divisor classes on every moduli space of stable complexes in the sense of Bridgeland. This divisor class varies naturally with the Bridgeland stability condition. For a generic stability condition on a K3 surface, we prove that this class is ample, thereby generalizing a result of Minamide, Yanagida, and Yoshioka. Our result also gives a systematic explanation of the relation between wall-crossing for Bridgeland-stability and the minimal model program for the moduli space. We give three applications of our method for classical moduli spaces of sheaves on a K3 surface: 1. We obtain a region in the ample cone in the moduli space of Gieseker-stable sheaves only depending on the lattice of the K3. 2. We determine the nef cone of the Hilbert scheme of n points on a K3 surface of Picard rank one when n is large compared to the genus. 3. We verify the "Hassett-Tschinkel/Huybrechts/Sawon" conjecture on the existence of a birational Lagrangian fibration for the Hilbert scheme in a new family of cases.