Writing equations of complex surface singularities from their topology
Jonathan Wahl
(University of North Carolina)

Via resolution, a singular point P of a normal complex surface X gives rise to a negative-definite configuration E of exceptional curves, and E is essentially equivalent to the local topology of X near P. Any E could occur for some X, and we are interested in the germ of X near P (i.e., the analytic local ring). Given E, can one write down equations for such an X? For all X? According to work of M. Artin in the 1960's, there are certain classes ("rational singularities"), recognizable from E, for which the data of E allows calculation of the Hilbert function at P: one knows the embedding dimension and the number of equations needed to define X--but not the explicit equations. In joint work with Walter Neumann, we study the (much) more general case that E is a tree of rational curves. Using an auxiliary "splice diagram", we show in many cases how to write down such X explicitly as the quotient of a complete intersection singularity by a finite abelian group. That is, we find the "universal abelian covering". As a special case, we show how to write down every rational surface singularity as an explicit abelian quotient of an explicit complete intersection. The starting point, in some sense, is reexamining the work of Felix Klein on rings of invariants of finite subgroups of SU(2).