Symmetries of Surface Singularities

The study of reductive group actions on a normal surface singularity X is facilitated by the fact that the group AutX of automorphisms of X has a maximal reductive algebraic subgroup G which contains every reductive algebraic subgroup of AutX up to conjugation. If X is not weighted homogeneous then this maximal group G is finite (Scheja, Wiebe). It has been determined for cusp singularities by Wall. On the other hand, if X is weighted homogeneous but not a cyclic quotient singularity then the connected component G1 of the unit coincides with the C defining the weighted homogeneous structure (Scheja, Wiebe and Wahl). Thus the main interest lies in the finite group G/G1. Not much is known about G/G1. Ganter has given a bound on its order valid for Gorenstein singularities which are not log-canonical. Aumann-Körber has determined G/G1 for all quotient singularities.