Extended McKay correspondence for quotient surface singularities

TL;DR

This paper extends the McKay correspondence to quotient surface singularities by explicitly describing the universal $G$-cluster sheaf and analyzing the quiver structure of the $G$-Hilbert scheme at each cluster, providing new insights into the geometry of these resolutions.

Contribution

It determines the generator sheaf of the universal $G$-cluster and studies the quiver structure of the $G$-Hilbert scheme for quotient surface singularities, strengthening the classical McKay correspondence.

Findings

Explicit description of the generator sheaf of the universal $G$-cluster.

Analysis of the quiver structure at each $G$-cluster.

Introduction of mono-special $O_{f A^2}$-submodules in the structure analysis.

Abstract

Let $G$ be a finite subgroup of $\mbox{GL}(2)$ acting on $\mathbf{A}^2\setminus\{0\}$ freely. The $G$-orbit Hilbert scheme $G\mbox{-Hilb}(\mathbf{A}^2)$ is a minimal resolution of the quotient $\mathbf{A}^2/G$. We determine the generator sheaf of the ideal defining the universal $G$-cluster over $G\mbox{-Hilb}(\mathbf{A}^2)$, which somewhat strengthens the well-known McKay correspondence for a finite subgroup of $\mbox{SL}(2)$. We also study the quiver structure of $G\mbox{-Hilb}(\mathbf{A}^2)$ at every $G$-cluster $O_{Z_y}=O_{\mathbf{A}^2}/I_y$ in terms of a collection of sort of minimal $G$-submodules of $O_{Z_y}$ (called mono-special $O_{\mathbf{A}^2}$-submodules) and generating $G$-submodules of $I_y$.