On rational homology disk smoothings of valency 4 surface singularities

TL;DR

This paper classifies and constructs rational homology disk smoothings for certain valency 4 surface singularities, providing explicit models, fundamental group descriptions, and a dimension formula, supporting the conjecture that only weighted homogeneous singularities admit such smoothings.

Contribution

It introduces a uniform quotient construction for QHD smoothings of valency 4 singularities, including a new class, and proves a dimension formula for QHD smoothing components.

Findings

Constructed explicit Q-Gorenstein smoothings for three classes of valency 4 singularities.

Proved the dimension of QHD smoothing components is 1 and smooth for valency 4 cases.

Most H-shaped resolution graphs do not admit QHD smoothings, supporting the conjecture.

Abstract

Thanks to the recent work of Bhupal, Stipsicz, Szabo, and the author, one has a complete list of resolution graphs of weighted homogeneous complex surface singularities admitting a rational homology disk ("QHD") smoothing, i.e., one with Milnor number 0. They fall into several classes, the most interesting of which are the three classes whose resolution dual graph has central vertex with valency 4. We give a uniform "quotient construction" of the QHD smoothings for these classes; it is an explicit Q-Gorenstein smoothing, yielding a precise description of the Milnor fibre and its non-abelian fundamental group. This had already been done for two of these classes in a previous paper; what is new here is the construction of the third class, which is far more difficult. In addition, we explain the existence of two different QHD smoothings for the first class. We also prove a general formula for the dimension of a QHD smoothing component for a rational surface singularity. A corollary is that for the valency 4 cases, such a component has dimension 1 and is smooth. Another corollary is that "most" H-shaped resolution graphs cannot be the graph of a singularity with a QHD smoothing. This result, plus recent work of Bhupal-Stipsicz, is evidence for a general Conjecture: The only complex surface singularities with a QHD smoothing are the (known) weighted homogeneous examples.