Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Resolutions and deformations of cyclic quotient surface singularities

Yukari Ito, Kohei Sato, Meral Tosun

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.

Resolutions and deformations of cyclic quotient surface singularities

Abstract

In this paper, we investigate the relations among various results concerning the minimal resolution of cyclic quotient singularities of the form . 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 -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.

Paper Structure

This paper contains 6 sections, 17 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. Group $\mathcal{C}_{n}$
  3. Group $\mathcal{C}_{n, q}$
  4. Toric construction
  5. McKay correspondence
  6. G-Hilbert scheme

Key Result

Proposition 2.1

An element $g^m$ with $m<n$ in $G$ has order $\frac{n}{(m, n)}$. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (33)

  • Proposition 2.1
  • Proposition 2.2
  • Definition 2.3
  • Theorem 3.1
  • Definition 3.2
  • Remark 3.3
  • Theorem 3.4
  • Proposition 3.5
  • Corollary 3.6
  • Corollary 3.7
  • ...and 23 more