© 1999 by London Mathematical Society
© The London Mathematical Society
Symmetries of Surface Singularities
Fachbereich Mathematik, Universität Mainz D 55099 Mainz, Germany mueller{at}mat.mathematik.uni-mainz.de
Received 24 October 1996. Revision received 9 June 1997.
The study of reductive group actions on a normal surface singularity X is facilitated by the fact that the group Aut X of automorphisms of X has a maximal reductive algebraic subgroup G which contains every reductive algebraic subgroup of Aut X 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, 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.
We propose to study G/G1 through the action of G on the minimal good resolution 
of X. If X is weighted homogeneous but not a cyclic quotient singularity, let E0 be the central curve of the exceptional divisor of . We show that the natural homomorphism G
Aut E0 has kernel C* and finite image. In particular, this re-proves the rest of Scheja, Wiebe and Wahl mentioned above. Moreover, it allows us to view G/G1 as a subgroup of Aut E0. For simple elliptic singularities it equals (ZbxZb)
Aut0 E0 where –b is the self-intersection number of E0, ZbxZb is the group of b-torsion points of the elliptic curve E0 acting by translations, and Aut0 E0 is the group of automorphisms fixing the zero element of E0. If E0 is rational then G/G1 is the group of automorphisms of E0 which permute the intersection points with the branches of the exceptional divisor while preserving the Seifert invariants of these branches. When there are exactly three branches we conclude that G/G1 is isomorphic to the group of automorphisms of the weighted resolution graph. This applies to all non-cyclic quotient singularities as well as to triangle singularities. We also investigate whether the maximal reductive automorphism group is a direct product G
G1xG/G1. This is the case, for instance, if the central curve E0 is rational of even self-intersection number or if X is Gorenstein such that its nowhere-zero 2-form 
has degree ±1. In the latter case there is a ‘natural’ section G/G1
G of GG/G1 given by the group of automorphisms in G which fix . For a simple elliptic singularity one has GG1xG/G1 if and only if –E0 · E0 = 1.