Resolutions and deformations of cyclic quotient surface singularities

TL;DR

This paper provides a comprehensive survey of the minimal resolution, deformations, and related structures of cyclic quotient surface singularities, unifying classical results with examples.

Contribution

It offers a unified exposition of key results on cyclic quotient singularities, connecting various approaches and inspiring potential generalizations.

Findings

Unified presentation of classical results and examples

Connections among resolutions, deformations, and McKay correspondence

Insights for extending to non-cyclic and higher-dimensional cases

Abstract

In this paper, we investigate the relations among various results concerning the minimal resolution of cyclic quotient singularities of the form $\mathbb{C}^2/G$. We refer to these as "bamboo-type" singularities, since the dual graphs of the exceptional curves in their resolutions resemble the shape of bamboo. We present classical results on the minimal resolution of singularities, the $G$-Hilbert scheme, the generalized McKay correspondence, deformations of singularities, and quiver varieties. These results have been obtained independently in different contexts, and here we provide a unified exposition enriched with numerous examples, which we hope will serve as a useful guide to the study of two-dimensional cyclic singularities. Moreover, this survey aims to offer insights that may inspire generalizations to non-cyclic singularities and to higher-dimensional quotient singularities.