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Projectivity and birational geometry of Bridgeland moduli spaces

Authors: Arend Bayer and Emanuele Macrì
Journal: J. Amer. Math. Soc. 27 (2014), 707-752
MSC (2010): Primary 14D20; Secondary 18E30, 14J28, 14E30
Published electronically: April 3, 2014
MathSciNet review: 3194493
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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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Additional Information

Arend Bayer
Affiliation: Department of Mathematics, University of Connecticut U-3009, 196 Auditorium Road, Storrs, Connecticut 06269-3009
Address at time of publication: School of Mathematics, The University of Edinburgh, James Clerk Maxwell Building, The King’s Buildings, Mayfield Road, Edinburgh, Scotland EH9 3JZ, United Kingdom
MR Author ID: 728427

Emanuele Macrì
Affiliation: Department of Mathematics, The Ohio State University, 231 W 18th Avenue, Columbus, Ohio 43210-1174

Keywords: Bridgeland stability conditions, derived category, moduli spaces of complexes, Mumford-Thaddeus principle
Received by editor(s): March 23, 2012
Received by editor(s) in revised form: April 8, 2013, and July 12, 2013
Published electronically: April 3, 2014
Article copyright: © Copyright 2014 American Mathematical Society