Minimal rational graphs admitting a QHD smoothing

Márton Beke

doi:10.2140/pjm.2026.344.1

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arXiv · 2504.06929

Minimal rational graphs admitting a QHD smoothing

Márton Beke

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Abstract

Using the picture deformation technique of De Jong-Van Straten we show that no singularity whose resolution graph has 3 or 4 large nodes, i.e., nodes satisfying d(v)+e(v)\leq -2, has a QHD smoothing. This is achieved by providing a general reduction algorithm for graphs with QHD smoothings, and enumeration. New examples and families are presented, which admit a combinatorial QHD 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 QHD smoothings with this method by using cusp singularities.

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Márton Beke. 2025-04-09. Minimal rational graphs admitting a QHD smoothing. https://doi.org/10.2140/pjm.2026.344.1

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