Projectivity and birational geometry of Bridgeland moduli spaces

By Arend Bayer, Emanuele Macrì. 2012

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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 wall-crossing for Bridgeland-stability 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 Gieseker-stable 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 “Hassett-Tschinkel/Huybrechts/Sawon” conjecture on the existence of a birational Lagrangian fibration for the Hilbert scheme in a new family of cases.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/eb0a4813be83a3256627aeeaf87b7a31c4cb1df2

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Citations in the corpus

The generalized Franchetta conjecture for some hyper-Kähler varieties, II (2020, background): “2 .2] using [BM14b].”
Hodge theory and Lagrangian planes on generalized Kummer fourfolds (2010, background): “if and only if : A[2] → Z/2Z is affine linear.”
Lectures on Bridgeland Stability (2016, methodology): “en a stability condition σ and a ﬁxed set of invariants v ∈Λ we will demonstrate how to construct a nef divisor on moduli spaces of σ-semistable objects with class v. This was originally described in [BM14b]. Assume that σ = (Z,A) is a stability condition on Db(X), S is a proper algebraic space of ﬁ”
Stability conditions on Kuznetsov components (2017, background): “greeshiftfunctor (1),thenwecanassumethatallobjects Φ(FX,L X )andFY,L Y havethesame numericalclassuptocomposing Φwiththeshiftfunctor [1]. Hence,wededuceanisomorphism between F(X) and F(Y). Finally, by [BM14a], moduli spaces of Bridgeland stable objects come equipped with a natural line bundle. If we ”
Stability conditions and extremal contractions (2012, background): “re several recent works relating Bridgeland stability conditions and MMP. In [2], the MMP of the Hilbert scheme of points in P2 is realized as a variation of Bridgeland stability conditions on P2. In [4], the moduli spaces of generic Bridgeland stability conditions on K3 surfaces are shown to be pro”
Birational models of moduli spaces of coherent sheaves on the projective plane (2016, background): “The birational geometry of moduli spaces of sheaves on surfaces has been studied a lot in recent years; see for instance Arcara, Bertram, Coskun and Huizenga [2], Bayer and Macrì [6; 7], Bertram, Martinez and Wang [9], Coskun and Huizenga [13; 14; 12], Coskun, Huizenga and Woolf [15], Li and Zhao [2”
Lectures on K3 Surfaces (2016, background): “See [264] for some comments and references and the recent preprints [40, 41, 637].”