Universal Abelian covers for surface singularities {zn=f(x,y)}

In recent work, W. D. Neumann and J. Wahl construct explicit equations for many interesting normal surface singularities with rational homology sphere links, which they call splice quotients. The construction begins with the topological type of a normal surface singularity, that is, a good resolution graph ¡ that is a tree of rational curves. If ¡ satisfies certain combinatorial conditions, then there exist splice quotients with resolution graph ¡. Let {zn = f(x, y)} define a surface Xf,n with an isolated singularity at the origin in C3. For f irreducible, we completely characterize, in terms of n and the Puiseux pairs of f, those Xf,n for which the resolution graph satisfies the combinatorial conditions defined by Neumann and Wahl. Briefly stated, we find that the conditions are not often satisfied. Furthermore, given a splice quotient (X, 0), it turns out that "equisingular deformations" of (X, 0) are usually not splice quotients, as we demonstrate already for singularities of the form {z2 = xP + yQ} with rational homology sphere link.