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MMP for moduli of sheaves on K3s via wall-crossing: nef and movable cones, Lagrangian fibrations

Arend Bayer, Emanuele Macrì

Number 20
Author Emanuele Macrì
Year 2013

We use wall-crossing with respect to Bridgeland stability conditions to systematically study the birational geometry of a moduli space M of stable sheaves on a K3 surface X:
1. We describe the nef cone, the movable cone, and the effective cone of M in terms of the Mukai lattice of X.
2. We establish a long-standing conjecture that predicts the existence of a birational Lagrangian fibration on M whenever M admits an integral divisor class D of square zero (with respect to the Beauville-Bogomolov form).
These results are proved using a natural map from the space of Bridgeland stability conditions Stab(X) to the cone Mov(X) of movable divisors on M; this map relates wall-crossing in Stab(X) to birational transformations of M. In particular, every minimal model of M appears as a moduli space of Bridgeland-stable objects on X.

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