Cyclic $n$-gonal Surfaces

TL;DR

This paper surveys the current research on automorphism groups of cyclic n-gonal surfaces, highlighting computational methods and providing explicit examples, with much of the content being expository and preparatory for future work.

Contribution

It offers an overview of ongoing research and computational techniques related to automorphism groups of cyclic n-gonal surfaces, with new explicit examples included.

Findings

Compilation of explicit examples of cyclic n-gonal surfaces

Description of computational methods for studying automorphisms

Overview of ongoing research and future directions

Abstract

A cyclic $n$-gonal surface is a compact Riemann surface $X$ of genus $g\geq 2$ admitting a cyclic group of conformal automorphisms $C$ of order $n$ such that the quotient space $X/C$ has genus 0. In this paper, we provide an overview of ongoing research into automorphism groups of cyclic $n$-gonal surfaces. Much of the paper is expository or will appear in forthcoming papers, so proofs are usually omitted. Numerous explicit examples are presented illustrating the computational methods currently being used to study these surfaces.