# The Minimal Model Program for moduli spaces of sheaves on K3 surfaces (Macrì)

In these lectures I will present joint work with Arend Bayer

(arXiv:1203.4613 and arXiv:1301.6968).

Topics include:

(1) The Positivity Lemma. Given a Bridgeland stability condition, I

will motivate and explain how to naturally associate a nef divisor on

the moduli space of Bridgeland stable objects.

(2) Explicit wall-crossing for K3 surfaces. I will explain

projectivity of moduli spaces of stable objects for K3s, and relate

concrete wall-crossing to concrete geometry (e.g., Brill-Noether loci,

relation to existence of g^1_n on curves in the case of the Hilbert

scheme).

(3) Systematic wall-crossing analysis for K3 surfaces in terms of its

lattice and applications. I will describe of the nef cone of the

moduli spaces in terms of the K3 lattice.

If time permits, I will also talk about Lagrangian fibrations, and

more systematic examples.