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

Abstract

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:(a) We describe the nef cone, the movable cone, and the effective cone of M in terms of the Mukai lattice of X. (b) 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.

Lectures on Bridgeland Stability

“…In particular cases, Theorem 8.6 can be be made more precise and general. In the case of the projective plane [CHW14,CH15,LZ16] and of K3 surfaces [BM14a,BM14b], given any primitive vector, varying stability conditions corresponds to a directed Minimal Model Program for the corresponding moduli space. This allows to completely describe the nef cone, the movable cone, and the pseudo-effective cone for them.…”

“…The study of the abelian surfaces case started in [MM13] and was completed in [MYY14,YY14]. The case of K3 surfaces was handled in [MYY14,BM14a,BM14b]. Enriques surfaces were studied in [Nue16].…”

Birational Geometry of the Moduli Space of Rank 2 Parabolic Vector Bundles on a Rational Curve

“…For each j, draw (1, 1) vector x j times and draw (1, −1) vector y j times. Then (3) and (4) imply that the path is lying on the region 0 ≤ y ≤ . By (5), the ending point is (|k|, 0).…”

“…For Hilbert scheme Hilb n (P 2 ) of n points on P 2 , many of its birational models appearing in Mori's program are moduli spaces of Bridgeland stable objects in D b (P 2 ) with certain stability condition ( [2]). For the moduli space of stable sheaves M H (v) on a K3 surface X, all flips of M H (v) are moduli spaces of Bridgeland stable objects in D b (X) ( [4]). …”

Rational Points on K3 Surfaces and Derived Equivalence

“…• the existence of vector bundles and twisted sheaves with prescribed invariants; • geometric interpretations of isogenies between K3 surfaces [Orl97,Căl00]; • the global Torelli theorem for holomorphic symplectic manifolds [Ver13,Huy12b]; • the analysis of stability conditions and its implications for birational geometry of moduli spaces of vector bundles and more general objects in the derived category [BMT14,BM14,Bri07].…”