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.

Readership

Advanced undergraduates, graduate students, and researchers.

Table of Contents

• Introduction
• Classification of finite subgroups of \(SL(3,\mathbb C)\)
• The invariant polynomials and their relations of linear groups of \(SL(3,\mathbb C)\)
• Gorenstein quotient singularities in dimension three