Minimal rational graphs admitting a $\mathbb{Q}$HD smoothing

Márton Beke

Vol. 344 (2026), No. 1, 1–25

DOI: 10.2140/pjm.2026.344.1

Abstract

Using the picture deformation technique of de Jong and van Straten we show that no singularity whose resolution graph has 3 or 4 large nodes, i.e., nodes satisfying $d(v)+e(v)\le -2$, has a $\mathbb{Q}\text{ HD}$ smoothing. This is achieved by providing a general reduction algorithm for graphs with $\mathbb{Q}\text{ HD}$ smoothings, and enumeration. New examples and families are presented, which admit a combinatorial $\mathbb{Q}\text{ HD}$ smoothing, i.e., the incidence relations for a sandwich presentation can be satisfied. We also give a new proof of the Bhupal–Stipsicz theorem on the classification of weighted homogeneous singularities admitting $\mathbb{Q}\text{ HD}$ smoothings with this method by using cusp singularities.

Keywords

rational homology disk, surface singularity, rational blowdown, Wahl's conjecture, smoothing, plane curve deformations

Mathematical Subject Classification

Primary: 14B07, 14J17, 32S25, 32S30, 51E30

Milestones

Received: 9 April 2025

Accepted: 14 May 2026

Published: 10 August 2026