Gorenstein Quotient Singularities in Dimension Three
Stephen Shing-Toung Yau · Yung Yu
Jan 1993 · American Mathematical Society: Memoirs of the American Mathematical Society Book 505 · American Mathematical Soc.
Ebook
88
Pages
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About this ebook
If $G$ is a finite subgroup of $GL(3,{\Bbb C )$, then $G$ acts on ${\Bbb C 3$, and it is known that ${\Bbb C 3/G$ is Gorenstein if and only if $G$ is a subgroup of $SL(3,{\Bbb C )$. In this work, the authors begin with a classification of finite subgroups of $SL(3,{\Bbb C )$, inlcuding 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 $GL(3,{\Bbb C )$. 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 ${\Bbb C 3/G$ 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.
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