Projectivity and birational geometry of Bridgeland moduli spaces

Bayer, Arend; Macrì, Emanuele

doi:10.1090/S0894-0347-2014-00790-6

Projectivity and birational geometry of Bridgeland moduli spaces
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by Arend Bayer and Emanuele Macrì

J. Amer. Math. Soc. 27 (2014), 707-752

DOI: https://doi.org/10.1090/S0894-0347-2014-00790-6

Published electronically: April 3, 2014

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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.

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