Resolution graphs of some surface singularities II. (generalized Iomdin series)

Abstract

In this article we give a construction of the resolution graphs of hypersurface surface singularities (X_k,0) given by generalized Iomdin series. All these resolution graphs are coordinated by an ``universal bi-colored graph'' which is associated with the ICIS determining the Iomdin series. The definition of this new graph is rather involved, and in concrete examples it is difficult to compute. Nevertheless, we present a large number of examples. This is very helpful in the exemplification of its properties as well. We then present a construction of the resolution graphs of the surface singularities above which uses the "universal bi-coloured graph" and the integer k. This is formulated in a purely combinatorial algorithm. The result is a highly non-trivial generalization of the case of cyclic coverings of smooth surfaces.