A generalisation of the pencil of Kuribayashi-Komiya quartics

TL;DR

This paper generalizes the Kuribayashi-Komiya quartic pencil to higher prime degrees, determining automorphism groups of the resulting Riemann surfaces and classifying surfaces with such automorphism groups.

Contribution

It introduces a family of generalized Kuribayashi-Komiya curves for primes p ≥ 5 and characterizes their automorphism groups, extending previous studies of the original quartic pencil.

Findings

Determined the full automorphism group for each smooth member of the generalized pencil.

Proved that no member of the generalized pencil is hyperelliptic.

Classified Riemann surfaces with automorphism groups isomorphic to the generic members' automorphism groups.

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 $\mathcal{F}_p.$