# Hyperelliptic quotients of generalized Humbert curves

## TL;DR

This paper characterizes subgroups of generalized Humbert groups acting freely on curves, identifying conditions under which the quotient surfaces are hyperelliptic, thus advancing understanding of the structure of these algebraic curves.

## Contribution

It provides a detailed description of subgroups leading to hyperelliptic quotients of generalized Humbert curves, a topic not previously fully explored.

## Key findings

- Identifies subgroups K of H with free action on S
- Determines when S/K is hyperelliptic
- Enhances understanding of quotient structures of Humbert curves

## Abstract

A group $H \cong {\mathbb Z}_{2}^{n}$, $n \geq 3$, of conformal automorphisms of a closed Riemann surface $S$ such that $S/H$ has genus zero and exactly $(n+1)$ cone points is called a generalized Humbert group of type $n$, in which case, $S$ is called a generalized Humbert curve of type $n$. It is known that a generalized Humbert curve $S$ of type $n \geq 4$ is non-hyperelliptic and that it admits a unique generalized Humbert group $H$ of type $n$. We describe those subgroups $K$ of $H$, acting freely on $S$, such that $S/K$ is hyperelliptic.