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Projectivity and Birational Geometry of Bridgeland moduli spaces

Arend Bayer, Emanuele Macri

Number 4
Author Dr. Emanuele Macrì
Year 2012

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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