# Gorenstein Quotient Singularities in Dimension Three

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*Stephen S.-T. Yau; Yung Yu*

If \(G\) is a finite subgroup of \(GL(3,{\mathbb C})\), then \(G\) acts on \({\mathbb C}^3\), and it is known that \({\mathbb C}^3/G\) is Gorenstein if and only if \(G\) is a subgroup of \(SL(3,{\mathbb C})\). In this work, the authors begin with a classification of finite subgroups of \(SL(3,{\mathbb C})\), including two types, (J) and (K), which have often been overlooked. They go on to present a general method for finding invariant polynomials and their relations to finite subgroups of \(GL(3,{\mathbb C})\). The method is, in practice, substantially better than the classical method due to Noether. Some properties of quotient varieties are presented, along with a proof that \({\mathbb C}^3/G\) has isolated singularities if and only if \(G\) is abelian and 1 is not an eigenvalue of \(g\) for every nontrivial \(g \in G\). The authors also find minimal quotient generators of the ring of invariant polynomials and relations among them.

#### Table of Contents

# Table of Contents

## Gorenstein Quotient Singularities in Dimension Three

- TABLE OF CONTENTS vii8 free
- CHAPTER 0 INTRODUCTION 110 free
- CHAPTER 1 CLASSIFICATION OF FINITE SUBGROUPS OF SL(3, C) 1019
- 1.1 Definitions 1019
- 1.2 Intransitive and imprimitive groups 1120
- 1.3 Remarks on the invariants of the groups (C) and (D) 1625
- 1.4 Groups having normal intransitive subgroups 1827
- 1.5 Primitive groups having normal imprimitive subgroups 1827
- 1.6 Primitive groups which are simple 2029
- 1.7 Primitive groups having normal intransitive subgroups (continued) 3443
- 1.8 Primitive groups having normal primitive subgroups 3544

- CHAPTER 2 THE INVARIANT POLYNOMIALS AND THEIR RELATIONS OF LINEAR GROUPS OF SL(3, C) 3847
- 2.1 Theorems 3948
- 2.2 The invariants of group of type (A) 4352
- 2.3 The invariants of group of type (B) 4756
- 2.4 The invariants of group of type (C) 6271
- 2.5 The invariants of group of type (D) 6473
- 2.6 The invariants of group (E) 6675
- 2.7 The invariants of group (F) 6776
- 2.8 The invariants of group (G) 6978
- 2.9 The invariants of group (H) 7281
- 2.10 The invariants of group (I) 7483
- 2.11 The invariants of group (J) 7685
- 2.12 The invariants of group (K) 7786
- 2.13 The invariants of group (L) 7887

- CHAPTER 3 GORENSTEIN QUOTIENT SINGULARITIES IN DIMENSION THREE 8291