Symmetries of Surface Singularities

The study of reductive group actions on a normal surface singularityX is facilitated by the fact that the group Aut X of automorphismsof X has a maximal reductive algebraic subgroup G which containsevery reductive algebraic subgroup of Aut X up to conjugation.If X is not weighted homogeneous then this maximal group G isfinite (Scheja, Wiebe). It has been determined for cusp singularitiesby Wall. On the other hand, if X is weighted homogeneous butnot a cyclic quotient singularity then the connected componentG₁ of the unit coincides with the C* defining the weighted homogeneousstructure (Scheja, Wiebe, Wahl). Thus the main interest liesin the finite group G/G₁. Not much is known about G/G₁. Ganterhas given a bound on its order valid for Gorenstein singularitieswhich are not log-canonical. Aumann-Körber has determinedG/G₁ for all quotient singularities. We propose to study G/G₁ through the action of G on the minimalgood resolution of X. If X is weightedhomogeneous but not a cyclic quotient singularity, let E₀ bethe central curve of the exceptional divisor of . We show that the natural homomorphism GAut E₀ haskernel C* and finite image. In particular, this re-proves therest of Scheja, Wiebe and Wahl mentioned above. Moreover, itallows us to view G/G₁ as a subgroup of Aut E₀. For simple ellipticsingularities it equals (Z_bxZ_b)Aut₀ E₀ where –b is theself-intersection number of E₀, Z_bxZ_b is the group of b-torsionpoints of the elliptic curve E₀ acting by translations, andAut₀ E₀ is the group of automorphisms fixing the zero elementof E₀. If E₀ is rational then G/G₁ is the group of automorphismsof E₀ which permute the intersection points with the branchesof the exceptional divisor while preserving the Seifert invariantsof these branches. When there are exactly three branches weconclude that G/G₁ is isomorphic to the group of automorphismsof the weighted resolution graph. This applies to all non-cyclicquotient singularities as well as to triangle singularities.We also investigate whether the maximal reductive automorphismgroup is a direct product GG₁xG/G₁. This is the case, for instance,if the central curve E₀ is rational of even self-intersectionnumber 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/G₁G of GG/G₁ given by the group of automorphisms inG which fix . For a simple elliptic singularity one has GG₁xG/G₁if and only if –E₀ · E₀ = 1.