Rational blow-downs and smoothings of surface singularities

TL;DR

This paper establishes a combinatorial criterion for identifying plumbing trees suitable for rational blow-downs and surface singularities with rational homology disk smoothings, introducing new examples including those with valency four nodes.

Contribution

It provides a necessary combinatorial condition for plumbing trees to admit rational blow-downs and presents new surface singularity examples with rational homology disk smoothings.

Findings

Derived a combinatorial condition for rational blow-down suitability

Identified new surface singularities with rational homology disk smoothings

Included examples with valency four nodes in resolution graphs

Abstract

In this paper we give a necessary combinatorial condition for a negative--definite plumbing tree to be suitable for rational blow--down, or to be the graph of a complex surface singularity which admits a rational homology disk smoothing. New examples of surface singularities with rational homology disk smoothings are also presented; these include singularities with resolution graph having valency four nodes.