Topology, geometry, and equations of normal surface singularities

This expository talk is an expanded version of a lecture at G.-M. Greuel's 60th Birthday Conference in Kaiserslautern in October, 2004. We survey recent work of Neumann-Wahl and others on the relation between topology and geometry of normal surface singularities, especially how to write explicit equations of a singularity with given topology, in case the link is a rational homology sphere (i.e., the resolution graph is a tree of rational curves). Specifically, under some reasonable conditions on the link, there is an explicit "complete intersection of splice type" whose link is the universal abelian cover of the original one, and for which the action of the covering group can be explicitly described. Recent work of T. Okuma shows every rational surface singularity can be described in this way. Full details are found in papers in Geometry and Topology, math.AG/0301165 and math.AG/0407287.