$G$-constellations and the maximal resolution of a quotient surface singularity

TL;DR

This paper characterizes when certain moduli spaces of $G$-constellations provide the maximal resolution of quotient surface singularities, especially for abelian or small groups, linking geometric resolutions with stability conditions.

Contribution

It establishes that a resolution of $\

Findings

Resolutions dominated by the maximal resolution correspond to generic stability parameters.

The moduli space of $G$-constellations can realize the maximal resolution for abelian or small groups.

Provides a criterion linking geometric resolutions with stability conditions in the moduli space.

Abstract

For a finite subgroup $G$ of $\operatorname{GL}(2, \mathbb C)$, we consider the moduli space ${\mathcal M}_\theta$ of $G$-constellations. It depends on the stability parameter $\theta$ and if $\theta$ is generic it is a resolution of singularities of $\mathbb C^2/G$. In this paper, we show that a resolution $Y$ of $\mathbb C^2/G$ is isomorphic to ${\mathcal M}_\theta$ for some generic $\theta$ if and only if $Y$ is dominated by the maximal resolution under the assumption that $G$ is abelian or small.