Complex surface singularities with rational homology disk smoothings

TL;DR

This paper classifies complex surface singularities with rational homology disk smoothings, completing the analytic classification and counting smoothings for all cases using rational curve configurations on surfaces.

Contribution

It completes the classification of singularities with rational homology disk smoothings, especially for the remaining types, and counts the smoothings in each case.

Findings

Unique QHD smoothing component in most cases

Complete analytic classification of these singularities

Method involves rational curve configurations on surfaces

Abstract

A cyclic quotient singularity of type $p^2/pq-1$ ($0<q<p, (p,q)=1$) has a smoothing whose Milnor fibre is a $\mathbb Q$HD, or rational homology disk (i.e., the Milnor number is $0$) ([9], 5.9.1). In the 1980's, we discovered additional examples of such singularities: three triply-infinite and six singly-infinite families, all weighted homogeneous. Later work of Stipsicz, Szab\'{o}, Bhupal, and the author ([7], [1]) proved that these were the only weighted homogeneous examples. In his UNC PhD thesis (unpublished but available at [2]), our student Jacob Fowler completed the analytic classification of these singularities, and counted the number of smoothings in each case, except for types $\mathcal W$, $\mathcal N$, and $\mathcal M$. In this paper, we describe his results, and settle these remaining cases; there is a unique $\mathbb Q$HD smoothing component except in the cases of an obvious symmetry of the resolution dual graph. The method involves study of configurations of rational curves on projective rational surfaces.