A generalisation of the pencil of Kuribayashi-Komiya quartics

Abstract

The pencil of Kuribayashi-Komiya quartics $$ x^4 + y^4 + z^4 + t(x^2y^2 + x^2z^2 + y^2z^2)=0 \, \mbox{ where } t \in \bar{\mathbb{C}} $$is a complex one-dimensional family of Riemann surfaces of genus three endowed with a group of automorphisms isomorphic to the symmetric group of order twenty-four. This pencil has been extensively studied from different points of view. This paper is aimed at studying, for each prime number $p \geqslant 5$, the pencil of \textit{generalised Kuribayashi-Komiya curves} $\mathcal{F}_p$, given by the curves $$x^{2p}+y^{2p}+z^{2p}+t(x^p y^p +x^p z^p +y^p z^p)=0\mbox{ where } t \in \bar{\mathbb{C}}.$$We determine the full automorphism group $G$ of each smooth member $X \in \mathcal{F}_p$ and study the action of $G$ and of its subgroups on $X$. In particular, we show that no member of the pencil is hyperelliptic. As a by-product, we derive a classification of all those Riemann surfaces of genus $(p-1)(2p-1)$ that are endowed with a group of automorphisms isomorphic to the full automorphism group of the generic smooth member of $\mathcal{F}_p.$