Symmetries of Surface Singularities

Details Symmetries of Surface Singularities Symmetries of Surface Singularities

1996-04-24

arxiv.org/abs/alg-geom/9604015v1

Mathematics

Algebraic Geometry

LaTeX, 24 pages, hard copies available

Scientific paper

The automorphism group ${\rm Aut}\: X$ of a weighted homogeneous normal surface singularity $X$ has a maximal reductive algebraic subgroup $G$ which contains every reductive algebraic subgroup of ${\rm Aut}\: X$ up to conjugation. In all cases except the cyclic quotient singularities the connected component $G_1$ of the unit equals ${\Bbb C}^*$. The induced action of $G$ on the minimal good resolution of $X$ embeds the finite group $G/G_1$ into the automorphism group of the central curve $E_0$ of the exceptional divisor. We describe $G/G_1$ as a subgroup of ${\rm Aut}\: E_0$ in case $E_0$ is rational as well as for simple elliptic singularities. Moreover, sufficient conditions for $G$ to be a direct product $G_1 \times G/G_1$ are presented. Finally, it is shown that $G/G_1$ acts faithfully on the integral homology of the link of $X$.

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