Smooth quotients of generalized Fermat curves

TL;DR

The paper studies generalized Fermat curves of type (p,n), describing their automorphism groups, explicit equations, and conditions under which their quotients are hyperelliptic, expanding understanding of their algebraic and geometric properties.

Contribution

It characterizes free subgroups of the automorphism group acting on these curves and provides explicit equations for their quotients, including hyperelliptic cases.

Findings

Explicit equations for quotients of generalized Fermat curves are derived.

Conditions for quotients to be hyperelliptic are determined.

Classification of free subgroups acting on these curves is provided.

Abstract

A closed Riemann surface $S$ is called a generalized Fermat curve of type $(p,n)$, where $n,p \geq 2$ are integers such that $(p-1)(n-1)>2$, if it admits a group $H \cong {\mathbb Z}_{p}^{n}$ of conformal automorphisms with quotient orbifold $S/H$ of genus zero with exactly $n+1$ cone points, each one of order $p$; in this case $H$ is called a generalized Fermat group of type $(p,n)$. In this case, it is known that $S$ is non-hyperelliptic and that $H$ is its unique generalized Fermat group of type $(p,n)$. Also, explicit equations for them, as a fiber product of classical Fermat curves of degree $p$, are known. For $p$ a prime integer, we describe those subgroups $K$ of $H$ acting freely on $S$, together with algebraic equations for $S/K$, and determine those $K$ such that $S/K$ is hyperelliptic.