Projectivity and Birational Geometry of Bridgeland 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.

v2: 50 pages, exposition substantially improved based on referee's comments. Accepted for publication in Journal of the AMS

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Cited by in corpus (11)

• A generalized Bogomolov-Gieseker inequality for the three-dimensional projective space
• Bridgeland's stability and the positive cone of the moduli spaces of stable objects on an abelian surface
• Mori cones of holomorphic symplectic varieties of K3 type
• The birational geometry of moduli space of sheaves on the projective plane
• ACM bundles on cubic fourfolds containing a plane
• Wall crossing of the moduli spaces of perverse coherent sheaves on a blow-up
• On $k$-gonal loci in Severi varieties on general $K3$ surfaces and rational curves on hyperkähler manifolds
• Stability conditions and extremal contractions
• Automorphisms of positive entropy on some hyperKahler manifolds via derived automorphisms of K3 surfaces
• Lagrangian embeddings of cubic fourfolds containing a plane
• Some remarks on Bridgeland stability conditions on K3 and Enriques surfaces