# Birational geometry of symplectic quotient singularities

Gwyn Bellamy, Alastair Craw

Research output: Contribution to journalArticle

### Abstract

For a finite subgroup \Gamma in SL(2, C) and for n ≥ 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 C2/\Gamma. It is well known that X := Hilb^[n](S) is a projective, crepant resolution
of the symplectic singularity C2n/\Gamma_n, where \Gamma_n = \Gamma ≀ S_n is the wreath product. We prove that every projective, crepant resolution of C2n/\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 C2n/\Gamma_n. As a corollary, we describe completely the ample and movable cones of X over C2n/\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.
Original language English 49 arXiv Submitted - 2018

### Cite this

Bellamy, G., & Craw, A. (2018). Birational geometry of symplectic quotient singularities. Manuscript submitted for publication.
In: arXiv, 2018.

Research output: Contribution to journalArticle

title = "Birational geometry of symplectic quotient singularities",
abstract = "For a finite subgroup \Gamma in SL(2, C) and for n ≥ 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 C2/\Gamma. It is well known that X := Hilb^[n](S) is a projective, crepant resolutionof the symplectic singularity C2n/\Gamma_n, where \Gamma_n = \Gamma ≀ S_n is the wreath product. We prove that every projective, crepant resolution of C2n/\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-stabilityconditions to birational transformations of X over C2n/\Gamma_n. As a corollary, we describe completely the ample and movable cones of X over C2n/\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.",
author = "Gwyn Bellamy and Alastair Craw",
year = "2018",
language = "English",
journal = "arXiv",

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AU - Craw, Alastair

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N2 - For a finite subgroup \Gamma in SL(2, C) and for n ≥ 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 C2/\Gamma. It is well known that X := Hilb^[n](S) is a projective, crepant resolutionof the symplectic singularity C2n/\Gamma_n, where \Gamma_n = \Gamma ≀ S_n is the wreath product. We prove that every projective, crepant resolution of C2n/\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-stabilityconditions to birational transformations of X over C2n/\Gamma_n. As a corollary, we describe completely the ample and movable cones of X over C2n/\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.

AB - For a finite subgroup \Gamma in SL(2, C) and for n ≥ 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 C2/\Gamma. It is well known that X := Hilb^[n](S) is a projective, crepant resolutionof the symplectic singularity C2n/\Gamma_n, where \Gamma_n = \Gamma ≀ S_n is the wreath product. We prove that every projective, crepant resolution of C2n/\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-stabilityconditions to birational transformations of X over C2n/\Gamma_n. As a corollary, we describe completely the ample and movable cones of X over C2n/\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.

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