Birational geometry of symplectic quotient singularities

For a finite subgroup $Γ\subset \mathrm{SL}(2,\mathbb{C})$ 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 $\mathbb{C}^2/Γ$. It is well known that $X:=\mathrm{Hilb}^{[n]}(S)$ is a projective, crepant resolution of the symplectic singularity $\mathbb{C}^{2n}/Γ_n$, where $Γ_n=Γ\wr\mathfrak{S}_n$ is the wreath product. We prove that every projective, crepant resolution of $\mathbb{C}^{2n}/Γ_n$ can be realised as the fine moduli space of $θ$-stable $Π$-modules for a fixed dimension vector, where $Π$ is the framed preprojective algebra of $Γ$ and $θ$ is a choice of generic stability condition. Our approach uses the linearisation map from GIT to relate wall crossing in the space of $θ$-stability conditions to birational transformations of $X$ over $\mathbb{C}^{2n}/Γ_n$. As a corollary, we describe completely the ample and movable cones of $X$ over $\mathbb{C}^{2n}/Γ_n$, 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 $Γ$ 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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