MMP for moduli of sheaves on K3s via wall-crossing: nef and movable cones, Lagrangian fibrations
By Arend Bayer, Emanuele Macrì. Inventiones Mathematicae, 2013
336 citations recorded by the source corpus.
We use wall-crossing with respect to Bridgeland stability conditions to systematically study the birational geometry of a moduli space $$M$$M of stable sheaves on a K3 surface $$X$$X: (a) We describe the nef cone, the movable cone, and the effective cone of $$M$$M in terms of the Mukai lattice of $$X$$X. (b) We establish a long-standing conjecture that predicts the existence of a birational Lagrangian fibration on $$M$$M whenever $$M$$M admits an integral divisor class $$D$$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 $$\mathop {\mathrm {Stab}}\nolimits (X)$$Stab(X) to the cone $$\mathop {{\mathrm {Mov}}}\nolimits (X)$$Mov(X) of movable divisors on $$M$$M; this map relates wall-crossing in $$\mathop {\mathrm {Stab}}\nolimits (X)$$Stab(X) to birational transformations of $$M$$M. In particular, every minimal model of $$M$$M appears as a moduli space of Bridgeland-stable objects on $$X$$X.
Subfields: Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/dd02fbdaa0a17fe64427baa1d450cf7f16a63813
How researchers see this paper
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In Geometry and Topology
- Benjamin Schmidt
- Chunyi Li
- Xiaolei Zhao
- Emanuele Macrì
- D. Huybrechts
- P. Stellari
- E. Markman
- Arend Bayer
Citations in the corpus
- Lectures on K3 Surfaces (2016, background): “See [264] for some comments and references and the recent preprints [40, 41, 637].”
- Projectivity and birational geometry of Bridgeland moduli spaces (2012, neutral): “We will treat this case in [7].”
- 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”
- A generalized Bogomolov–Gieseker inequality for the smooth quadric threefold (2013, background): “Examples of successful applications are found in the birational geometry of Hilbert schemes of points on smooth projective surfaces (see, for example, [2, 4, 15, 22]).”