Walls for $G$-Hilb via Reid's Recipe

Abstract

The three-dimensional McKay correspondence seeks to relate the geometry of crepant resolutions of Gorenstein $3$-fold quotient singularities ${\mathbb{A}}^3/G$ with the representation theory of the group $G$. The first crepant resolution studied in depth was the $G$-Hilbert scheme $G\text{-Hilb}{\mathbb{A}}^3$, which is also a moduli space of $\theta$-stable representations of the McKay quiver associated to $G$. As the stability parameter $\theta$ varies, we obtain many other crepant resolutions. In this paper we focus on the case where $G$ is abelian, and compute explicit inequalities for the chamber of the stability space defining $G\text{-Hilb}{\mathbb{A}}^3$ in terms of a marking of exceptional subvarieties of $G\text{-Hilb}{\mathbb{A}}^3$ called Reid's recipe. We further show which of these inequalities define walls. This procedure depends only on the combinatorics of the exceptional fibre and has applications to the birational geometry of other crepant resolutions.