A class of pseudoreal Riemann surfaces with diagonal automorphism group

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 $d\ge 4$ in ${\mathbb{P}}_{\mathbb{C}}^2$. We characterize pseudoreal-plane Riemann surfaces $\mathcal{S}$, whose conformal automorphism group ${\mathrm{Aut}}_{+}(\mathcal{S})$ is ${\mathrm{PGL}}_3(\mathbb{C})$-conjugate to a finite non-trivial group $G$ that leaves invariant infinitely many points of ${\mathbb{P}}_{\mathbb{C}}^2$. In particular, we show that such pseudoreal-plane Riemann surfaces exist only if ${\mathrm{Aut}}_{+}(\mathcal{S})$ 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 $n=d/p$.

14 pages