Gorenstein Quotient Singularities in Dimension Three (Memoirs of the American Mathematical Society)

If $G$ is a finite subgroup of $G\!L(3,{\mathbb C})$, 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 $S\!L(3,{\mathbb C})$. In this work, the authors begin with a classification of finite subgroups of $S\!L(3,{\mathbb C})$, 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 $G\!L(3,{\mathbb C})$. 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.

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.

In chapter one we address the classification of finite subgroups of [italic capitals]SL([bold]3,[double-struck capital]C). This is followed by a general method to find invariant polynomials and their relations of finite subgroups of [italic capitals]GL([bold]3,[double-struck capital]C). Lastly, we recall some properties of quotient varieties and prove that [double-struck capital]C3/[italic capital]G has isolated singularities if and only if [italic capital]G is abelian and 1 is not an eigenvalue of g in [italic capital]G If $G$ is a finite subgroup of $G\!L(3,{\mathbb C})$, 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 $S\!L(3,{\mathbb C})$. This book presents the classification of finite subgroups of $S\!L(3,{\mathbb C})$, including two types, (J) and (K).