Birational geometry of symplectic quotient singularities

For a finite subgroup $\mathrm{\Gamma }\subset \mathrm{S}\mathrm{L}(2,\mathbb{C})$ 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 ${\mathbb{C}}^2\mathrm{/}\mathrm{\Gamma }$. It is well known that $X:={\mathrm{H}\mathrm{i}\mathrm{l}\mathrm{b}}^{[n]}(S)$ is a projective, crepant resolution of the symplectic singularity ${\mathbb{C}}^{2n}\mathrm{/}{\mathrm{\Gamma }}_n$, where ${\mathrm{\Gamma }}_n=\mathrm{\Gamma }\wr {\mathfrak{S}}_n$ is the wreath product. We prove that every projective, crepant resolution of ${\mathbb{C}}^{2n}\mathrm{/}{\mathrm{\Gamma }}_n$ can be realised as the fine moduli space of $\theta$-stable $\mathrm{\Pi }$-modules for a fixed dimension vector, where $\mathrm{\Pi }$ is the framed preprojective algebra of $\mathrm{\Gamma }$ 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 ${\mathbb{C}}^{2n}\mathrm{/}{\mathrm{\Gamma }}_n$. As a corollary, we describe completely the ample and movable cones of $X$ over ${\mathbb{C}}^{2n}\mathrm{/}{\mathrm{\Gamma }}_n$, 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 $\mathrm{\Gamma }$ 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.