Birational geometry of symplectic quotient singularities

Abstract

For a finite subgroup $\Gamma\subset \mathrm{SL}(2,\mathbb{C})$ and for $n\geq 1$, 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 $X:=\mathrm{Hilb}^{[n]}(S)$ 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.