Generalized superelliptic Riemann surfaces
Ruben A. Hidalgo, Saúl Quispe, Tony Shaska
TL;DR
This work introduces generalized superelliptic curves as a natural extension of hyperelliptic curves, characterized by a central order $n$ automorphism whose quotient has genus zero. It develops an algebraic framework, via automorphism groups and Harvey-type data, to describe these curves and provides necessary and sufficient conditions for when a cyclic $n$-gonal automorphism is generalized superelliptic, including canonical models. A central result is the uniqueness of the generalized superelliptic group $H$ in all non-exceptional cases, with a precise exceptional family for even $n$ where non-uniqueness occurs; these findings yield criteria for definability over the field of moduli when the quotient by $H$ has suitably rich automorphism structure. The paper further offers constructive tools and appendices (Horiuchi-based methods and explicit computations) to derive explicit algebraic equations for generalized superelliptic curves and to classify cyclic $n$-gonal curves, facilitating practical computations and applications in moduli problems.
Abstract
A closed Riemann surface $\mathcal X$, of genus $g \geq 2$, is called a generalized superelliptic curve of level $n \geq 2$ if it admits an order $n$ conformal automorphism $τ$ so that $\mathcal X/\langle τ\rangle$ has genus zero and $τ$ is central in ${\rm Aut}(\mathcal X)$; the cyclic group $H=\langle τ\rangle$ is called a generalized superelliptic group of level $n$ for $\mathcal X$. These Riemann surfaces are natural generalizations of hyperelliptic Riemann surfaces (when $n=2$). We provide an algebraic curve description of these Riemann surfaces in terms of their groups of automorphisms. Also, we observe that the generalized superelliptic group $H$ of level $n$ is unique, with the exception of a very particular family of exceptional generalized superelliptic Riemann surfaces for $n$ even. In particular, the uniqueness holds if either: (i) $n$ is odd or (ii) the quotient $\mathcal X/H$ has all its cone points of order $n$ (for instance, when $\mathcal X$ is a superelliptic curve of level $n$). In the non-exceptional case, we use this uniqueness property of its generalized superelliptic group $H$ to observe that the corresponding curves are definable over their fields of moduli if ${\rm Aut}(\mathcal X)/H$ is neither trivial or cyclic.