A generalisation of the pencil of Kuribayashi-Komiya quartics

Abstract

The pencil of Kuribayashi-Komiya quartics$x^4+y^4+z^4+t(x^2{y}^2+x^2{z}^2+y^2{z}^2)=0\text{ where }t\in \overline{C}$ 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 $p⩾5$, the pencil of generalised Kuribayashi-Komiya curves $F_p$, given by the curves$x^{2p}+y^{2p}+z^{2p}+t(x^p{y}^p+x^p{z}^p+y^p{z}^p)=0\text{ where }t\in \overline{C}.$

We determine the full automorphism group G of each smooth member $X\in F_p$ 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 $F_p$.