Classical curves via one-vertex maps

TL;DR

This paper characterizes classical algebraic curves using one-vertex maps, classifying these maps by automorphism group size to understand their symmetries and connections to Galois actions.

Contribution

It introduces a new classification of one-vertex maps based on automorphism groups, linking them to well-known algebraic curves and their symmetries.

Findings

Characterization of algebraic curves via one-vertex maps

Classification of maps according to automorphism group size

Insights into the symmetries of classical curves

Abstract

One-vertex maps (a type of dessin d'enfant) give a uniform characterization of certain well-known algebraic curves, including those of Klein, Wiman, Accola-Maclachlan and Kulkarni. The characterization depends on a new classification of one-vertex (dually, one-face or unicellular) maps according to the size of the group of map automorphisms. We use an equivalence relation appropriate for studying the faithful action of the absolute Galois group on dessins, although we do not pursue that line of inquiry here.