A generalisation of the pencil of Kuribayashi-Komiya quartics

2025/07/03 by Vega, Valentina Moreno, Reyes-Carocca, Sebastián · 1 citation
#14H30 #14H37 #30F10 #30F35 #Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2507.03128

Abstract

The pencil of Kuribayashi-Komiya quartics x⁴ + y⁴ + z⁴ + t(x²y² + x²z² + y²z²)=0 where t ∈ ℂis 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 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 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)(2p-1) that are endowed with a group of automorphisms isomorphic to the full automorphism group of the generic smooth member of F_p.

Citations

• On quartics with the maximal number of the maximal tangency lines
• The Hurwitz curve over a finite field and its Weierstrass points for the morphism of lines
• Automorphisms of the Generalized Fermat curves
• Group actions on Jacobian varieties

Cited by

• \mathbb Z_k^m-actions of signature (0;k,\stackreln+1…,k)

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