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Gorenstein Quotient Singularities in Dimension Three

Stephen S.-T. Yau University of Illinois at Chicago
Yung Yu National Cheng Kung University
Available Formats:
Electronic ISBN: 978-1-4704-0082-8
Product Code: MEMO/105/505.E
List Price: $36.00 MAA Member Price:$32.40
AMS Member Price: $21.60 Click above image for expanded view Gorenstein Quotient Singularities in Dimension Three Stephen S.-T. Yau University of Illinois at Chicago Yung Yu National Cheng Kung University Available Formats:  Electronic ISBN: 978-1-4704-0082-8 Product Code: MEMO/105/505.E  List Price:$36.00 MAA Member Price: $32.40 AMS Member Price:$21.60
• Book Details

Memoirs of the American Mathematical Society
Volume: 1051993; 88 pp
MSC: Primary 14; 32;

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.

• Chapters
• 0. Introduction
• 1. Classification of finite subgroups of $SL(3, \mathbb {C})$
• 2. The invariant polynomials and their relations of linear groups of $SL(3, \mathbb {C})$
• 3. Gorenstein quotient singularities in dimension three
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Volume: 1051993; 88 pp
MSC: Primary 14; 32;

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.

• 1. Classification of finite subgroups of $SL(3, \mathbb {C})$
• 2. The invariant polynomials and their relations of linear groups of $SL(3, \mathbb {C})$