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

Yau, Stephen Shing-Toung; Yu, Yung

Softcover

ISBN 10: 0821825674 ISBN 13: 9780821825679

Publisher: Amer Mathematical Society, 1993

Synopsis

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.

Edition details

• Publisher: Amer Mathematical Society
• Publication date: 1993
• Language: English
• ISBN 10: 0821825674
• ISBN 13: 9780821825679
• Binding: Paperback