Birational geometry of symplectic quotient singularities

Bellamy, G. and Craw, A. (2020) Birational geometry of symplectic quotient singularities. Inventiones Mathematicae, 222(2), pp. 399-468. (doi: 10.1007/s00222-020-00972-9)

Abstract

For a finite subgroup \Gamma \subset \mathrm {SL}(2,\mathbb {C}) and for n\ge 1, we use variation of GIT quotient for Nakajima quiver varieties to study the birational geometry of the Hilbert scheme of n points on the minimal resolution S of the Kleinian singularity \mathbb {C}^2/\Gamma . It is well known that X:={{\,\mathrm{{\mathrm {Hilb}}}\,}}^{[n]}(S) is a projective, crepant resolution of the symplectic singularity \mathbb {C}^{2n}/\Gamma _n, where \Gamma _n=\Gamma \wr \mathfrak {S}_n is the wreath product. We prove that every projective, crepant resolution of \mathbb {C}^{2n}/\Gamma _n can be realised as the fine moduli space of \theta -stable \Pi -modules for a fixed dimension vector, where \Pi is the framed preprojective algebra of \Gamma and \theta is a choice of generic stability condition. Our approach uses the linearisation map from GIT to relate wall crossing in the space of \theta -stability conditions to birational transformations of X over \mathbb {C}^{2n}/\Gamma _n. As a corollary, we describe completely the ample and movable cones of X over \mathbb {C}^{2n}/\Gamma _n, and show that the Mori chamber decomposition of the movable cone is determined by an extended Catalan hyperplane arrangement of the ADE root system associated to \Gamma by the McKay correspondence. In the appendix, we show that morphisms of quiver varieties induced by variation of GIT quotient are semismall, generalising a result of Nakajima in the case where the quiver variety is smooth.

• Item Type: Articles
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Bellamy, Professor Gwyn
• Authors: Bellamy, G., and Craw, A.
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: Inventiones Mathematicae
• Publisher: Springer
• ISSN: 0020-9910
• ISSN (Online): 1432-1297
• Published Online: 30 April 2020
• Copyright Holders: Copyright © 2020 The Authors
• First Published: First published in Inventiones Mathematicae 222(2): 399-468
• Publisher Policy: Reproduced under a Creative Commons licence