Symmetries of Surface Singularities

Abstract

The automorphism group $\mathrm{Aut}X$ of a weighted homogeneous normal surface singularity $X$ has a maximal reductive algebraic subgroup $G$ which contains every reductive algebraic subgroup of $\mathrm{Aut}X$ up to conjugation. In all cases except the cyclic quotient singularities the connected component $G_1$ of the unit equals ${\mathbb{C}}^{\ast }$. 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 $\mathrm{Aut}{E}_0$ 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 $X$.

LaTeX, 24 pages, hard copies available