A generalisation of the pencil of Kuribayashi-Komiya quartics

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 where t ∈ 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 2 p + y 2 p + z 2 p + t ( x p y p + x p z p + y p z p ) = 0 where t ∈ C ¯ . We determine the full automorphism group G of each smooth member X ∈ 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 ) ( 2 p − 1 ) that are endowed with a group of automorphisms isomorphic to the full automorphism group of the generic smooth member of F p .