Resolution graphs of some surface singularities I. (cyclic coverings)

Abstract

The article starts with some introductory material about resolution graphs of normal surface singularities (definitions, topological/homological properties, etc). We then discuss the case when the normal surface singularity is an N-fold cyclic covering of a surface germ, branched along a curve given by the germ of an analytic function f. We present non-trivial examples in order to show that from the embedded resolution graph G of f in general it is not possible to recover the resolution graph of the cyclic covering. The main results are the construction of a ``universal covering graph'' from the topology of the germ f, and the completely combinatorial construction of the resolution graph of cyclic coverings from this universal graph of f and the integer N. For this we also prove some purely graph-theoretical classification theorems of ``covering graphs''. In the last part, we connect the properties of the universal covering graph with the topological invariants of f, e.g. with the nilpotent part of its algebraic monodromy.