Normal surface singularities with an integral homology sphere link related to space monomial curves with a plane semigroup

TL;DR

This paper studies a family of normal surface singularities with integral homology sphere links, related to space monomial curves with plane semigroups, using toric resolutions to analyze their topological properties.

Contribution

It introduces a new family of singularities linked to space monomial curves and applies toric resolution techniques to investigate their topological invariants.

Findings

The link of these singularities is an integral homology sphere.

A characterization based on intersection matrix determinants is used.

Partial toric resolutions facilitate the study of these singularities.

Abstract

In this article, we consider an infinite family of normal surface singularities with an integral homology sphere link which is related to the family of space monomial curves with a plane semigroup. These monomial curves appear as the special fibers of equisingular families of curves whose generic fibers are a complex plane branch, and the related surface singularities appear in a proof of the monodromy conjecture for these curves. To investigate whether the link of a normal surface singularity is an integral homology sphere, one can use a characterization that depends on the determinant of the intersection matrix of a (partial) resolution. To study our family, we apply this characterization with a partial toric resolution of our singularities constructed as a sequence of weighted blow-ups.