# Projectivity and Birational Geometry of Bridgeland moduli spaces

Arend Bayer, Emanuele Macri

Number | 4 |
---|---|

Author | Dr. Emanuele MacrĂ¬ |

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.