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MMP FOR MODULI OF SHEAVES ON K3S VIA WALLCROSSING: NEF AND MOVABLE CONES, LAGRANGIAN FIBRATIONS
"... ABSTRACT. We use wallcrossing 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) ..."
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Cited by 35 (2 self)
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ABSTRACT. We use wallcrossing 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 longstanding 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 BeauvilleBogomolov 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 wallcrossing in Stab(X) to birational transformations of M. In particular, every minimal model of M appears as a moduli space of Bridgelandstable objects on X. CONTENTS
Projectivity and birational geometry of Bridgeland moduli spaces
, 2012
"... ABSTRACT. 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 gene ..."
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Cited by 34 (2 self)
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ABSTRACT. 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 wallcrossing for Bridgelandstability 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 Giesekerstable 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 “HassettTschinkel/Huybrechts/Sawon ” conjecture on the existence of a birational Lagrangian fibration for the Hilbert scheme in a new family of cases.
THE MINIMAL MODEL PROGRAM FOR THE HILBERT SCHEME OF POINTS ON P 2 AND BRIDGELAND STABILITY
"... Abstract. In this paper, we study the birational geometry of the Hilbert scheme P 2[n] of npoints on P 2. We discuss the stable base locus decomposition of the effective cone and the corresponding birational models. We give modular interpretations to the models in terms of moduli spaces of Bridgela ..."
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Cited by 34 (10 self)
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Abstract. In this paper, we study the birational geometry of the Hilbert scheme P 2[n] of npoints on P 2. We discuss the stable base locus decomposition of the effective cone and the corresponding birational models. We give modular interpretations to the models in terms of moduli spaces of Bridgeland semistable objects. We construct these moduli spaces as moduli spaces of quiver representations using G.I.T. and thus show that they are projective. There is a precise correspondence between wallcrossings in the Bridgeland stability manifold and wallcrossings between Mori cones. For n ≤ 9, we explicitly determine the walls in both interpretations and describe the corresponding flips and divisorial contractions. Contents
The birational geometry of the Hilbert scheme of points on surfaces
"... In this paper, we study the birational geometry of the Hilbert scheme of points on a smooth, projective surface, with special emphasis on rational surfaces such as P2,P1 × P1 and F1. We discuss constructions of ample divisors and determine the ample cone for Hirzebruch surfaces and del Pezzo surfac ..."
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Cited by 12 (8 self)
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In this paper, we study the birational geometry of the Hilbert scheme of points on a smooth, projective surface, with special emphasis on rational surfaces such as P2,P1 × P1 and F1. We discuss constructions of ample divisors and determine the ample cone for Hirzebruch surfaces and del Pezzo surfaces with K2 ≥ 2. As a corollary, we show that the Hilbert scheme of points on a Fano surface is a Mori dream space. We then discuss effective divisors on Hilbert schemes of points on surfaces and determine the stable base locus decomposition completely in a number of examples. Finally, we interpret certain birational models as moduli spaces of Bridgeland stable objects. When the surface is P1 × P1 or F1, we find a precise correspondence between the Mori walls and the Bridgeland walls, extending the results of [ABCH] to these surfaces.
Effective divisors on the Hilbert scheme of points in the plane and interpolation for stable bundles
 J. Algebraic Geom
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Interpolation, Bridgeland stability and monomial schemes in the plane
 J. Math. Pures Appl
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Introduction to stability conditions
 Preprint (2011), arXiv:1111.1745. 66 Thomas Brüstle and Dong Yang
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