Classification of wormhole singularities

TL;DR

This paper classifies all wormhole singularities, a specific type of surface singularity, using a new combinatorial framework, and provides an alternative proof for a known theorem on extremal P-resolutions.

Contribution

It introduces a novel combinatorial method based on coherent graphs of framed triangulated polygons to classify wormhole singularities and proves a related extremal resolution theorem.

Findings

Complete classification of wormhole singularities

Development of a new combinatorial framework

Alternative proof of the maximum extremal P-resolutions theorem

Abstract

We classify all wormhole singularities, i.e. cyclic quotient surface singularities admitting at least two extremal P-resolutions, thereby solving an open problem posed by Urz\'ua. Our approach introduces a new combinatorial framework based on what we call the coherent graph of a framed triangulated polygon. As an application, we give an alternative proof of the Hacking-Tevelev-Urz\'ua theorem on the maximum number of extremal P-resolutions of a cyclic quotient singularity.