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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
Front Cover for Gorenstein Quotient Singularities in Dimension Three
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
Front Cover for Gorenstein Quotient Singularities in Dimension Three
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  • Front Cover for Gorenstein Quotient Singularities in Dimension Three
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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
  • 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.

    Readership

    Advanced undergraduates, graduate students, and researchers.

  • Table of Contents
     
     
    • 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
  • Requests
     
     
    Review Copy – for reviewers who would like to review an AMS book
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
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.

Readership

Advanced undergraduates, graduate students, and researchers.

  • 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
Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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