A class of pseudoreal Riemann surfaces with diagonal automorphism group

TL;DR

This paper characterizes a special class of pseudoreal Riemann surfaces with diagonal automorphism groups, providing explicit examples and conditions under which they exist, especially focusing on plane surfaces with cyclic automorphism groups.

Contribution

It introduces a characterization of pseudoreal-plane Riemann surfaces with specific automorphism groups and constructs explicit examples for degrees of the form 2pm with odd m.

Findings

Existence of pseudoreal-plane Riemann surfaces with cyclic automorphism groups of even order dividing the degree.

Such surfaces exist only if the automorphism group is cyclic of even order dividing the degree.

Explicit examples are constructed for degrees of the form 2pm with m>1 odd, p prime, and automorphism order n=d/p.

Abstract

A Riemann surface $\mathcal{S}$ having field of moduli $\mathbb{R}$, but not a field of definition, is called \emph{pseudoreal}. This means that $\mathcal{S}$ has anticonformal automorphisms, but non of them is an involution. We call a Riemann surface $\mathcal{S}$ \emph{plane} if it can be described by a smooth plane model of some degree $d\geq4$ in $\mathbb{P}^2_{\mathbb{C}}$. We characterize pseudoreal-plane Riemann surfaces $\mathcal{S}$, whose conformal automorphism group $\operatorname{Aut}_+(\mathcal{S})$ is $\operatorname{PGL}_3(\mathbb{C})$-conjugate to a finite non-trivial group $G$ that leaves invariant infinitely many points of $\mathbb{P}^2_{\mathbb{C}}$. In particular, we show that such pseudoreal-plane Riemann surfaces exist only if $\operatorname{Aut}_+(\mathcal{S})$ is cyclic of even order $n$ dividing the degree $d$. Explicit examples are given, for any degree $d=2pm$ with $m>1$ odd, $p$ is prime and $n=d/p$.