SYMMETRIES OF SURFACE SINGULARITIES

Abstract

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 [Copf ]* 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 X˜ of X. If X is weighted homogeneous but not a cyclic quotient singularity, let E₀ be the central curve of the exceptional divisor of X˜. We show that the natural homomorphism G→Aut E₀ has kernel [Copf ]* 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 (ℤ_b×ℤ_b)[rtimes ]Aut₀E₀ where −b is the self-intersection number of E₀, ℤ_b×ℤ_b is the group of b-torsion points of the elliptic curve E₀ acting by translations, and Aut₀E₀ 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≃G₁×G/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 ω has degree ±1. In the latter case there is a ‘natural’ section G/G₁[rarrhk ]G of G[rarrhk ]G/G₁ given by the group of automorphisms in G which fix ω. For a simple elliptic singularity one has G≃G₁×G/G₁ if and only if −E₀ · E₀ = 1.