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

## TL;DR

This paper explicitly describes the stability conditions defining the $G$-Hilbert scheme in the context of the McKay correspondence for abelian groups, linking combinatorics of exceptional fibers to birational geometry.

## Contribution

It provides explicit inequalities for the stability chamber of $G$-Hilb in terms of Reid's recipe, clarifying the wall-crossing behavior for abelian groups.

## Key findings

- Derived inequalities for the stability chamber of $G$-Hilb.
- Identified which inequalities define walls in the stability space.
- Connected combinatorics of exceptional fibers to birational geometry applications.

## 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.