Almost-Regular Dessins on a Sphere and Torus

TL;DR

This paper investigates the realizability of certain ramification data on spheres and tori using dessins d'enfant, confirming most cases and identifying exceptions, thus advancing understanding of the Hurwitz problem.

Contribution

It provides a classification of genus 0 and 1 ramification data realizability using dessins d'enfant, including new results and exceptions.

Findings

Most small changes in genus 1 ramification data are realizable.

Four specific families of genus 1 data are nonrealizable.

A similar classification is achieved for genus 0 data.

Abstract

The Hurwitz problem asks which ramification data are realizable, that is appear as the ramification type of a covering. We use dessins d'enfant to show that families of genus 1 regular ramification data with small changes are realizable with the exception of four families which were recently shown to be nonrealizable. A similar description holds in the case of genus 0 ramification data.