Gorenstein Quotient Singularities in Dimension Three Paperback / softback

by Stephen Shing-Taung Yau, Yu Yung

Part of the Memoirs of the American Mathematical Society series

Paperback / softback

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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.