Topology, Geometry, and Equations of Normal Surface Singularities

In continuing joint work with Walter Neumann, we consider the relationship between three different points of view in describing a (germ of a) complex normal surface singularity. The explicit equations of a singularity allow one to talk about hypersurfaces, complete intersections, weighted homogeneity, Hilbert function, etc. The geometry of the singularity could involve analytic aspects of a good resolution, or existence and properties of Milnor fibres; one speaks of geometric genus, Milnor number, rational singularities, the Gorenstein and Q-Gorenstein properties, etc. The topology of the singularity means the description of its link, or equivalently (by a theorem of Neumann) the configuration of the exceptional curves in a resolution. We survey ongoing work ([15],[16]) with Neumann to study the possible geometry and equations when the topology of the link is particularly simple, i.e. the link has no rational homology, or equivalently the exceptional configuration in a resolution is a tree of rational curves. Given such a link, we ask whether there exist “nice” singularities with this topology. In our situation, that would ask if the singularity is a quotient of a special kind of explicitly given complete intersection (said to be “of splice type”) by an explicitly given abelian group; on the topological level, this quotient gives the universal abelian cover of the link. Our major result gives a topological condition (i.e., a condition on the resolution graph) that there exists a singularity which arises in this way (and hence one whose equations can be written “explicitly”). T. Okuma ([18]) has recently proved our Conjecture that rational and minimally elliptic singularities are all “splice-quotients”. We summarize first the well-studied case of plane curve singularities, to see what one might mean about geometry, topology, and equations in that case. There follows an introductory discussion of normal surface singularities, before considering our recent work. The purpose of the article is to survey the main ideas and directions, rather than to describe details, which can be found in other papers such as [15].