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Emanuele Macri : MMP for moduli spaces of sheaves on K3 surfaces and Cone Conjectures

We report on joint work with A. Bayer on how one can use wall-crossing techniques to study the birational geometry of a moduli space M of Gieseker-stable sheaves on a K3 surface X. In particular: (--) We will give a "modular interpretation" for all minimal models of M. (--) We will describe the nef cone, the movable cone, and the effective cone of M in terms of the algebraic Mukai lattice of X. (--) We will establish the so called Tyurin/Bogomolov/Hassett-Tschinkel/Huybrechts/Sawon Conjecture on the existence of Lagrangian fibrations on M.

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