On the topology of surface singularities {z^n=f(x,y)}, for f irreducible

TL;DR

This paper characterizes when surface singularities defined by z^n=f(x,y) with f irreducible have resolution graphs satisfying conditions for splice quotients, linking topological properties to combinatorial data of f.

Contribution

It provides a complete topological characterization of such singularities based on n and Puiseux pairs of f, clarifying when they meet splice quotient conditions.

Findings

Resolved the topological criteria for surface singularities to be splice quotients.

Connected resolution graph properties with algebraic data of f.

Enhanced understanding of the topology of surface singularities with irreducible f.

Abstract

The splice quotients are an interesting class of normal surface singularities with rational homology sphere links, defined by W. Neumann and J. Wahl. If Gamma is a tree of rational curves that satisfies certain combinatorial conditions, then there exist splice quotients with resolution graph Gamma. Suppose the equation z^n=f(x,y) defines a surface X_{f,n} with an isolated singularity at the origin in C^3. For f irreducible, we completely characterize, in terms of n and a variant of the Puiseux pairs of f, those X_{f,n} for which the resolution graph satisfies the combinatorial conditions that are necessary for splice quotients. This result is topological; whether or not X_{f,n} is analytically isomorphic to a splice quotient is treated separately.