Classification of 1-Weierstrass points on Kuribayashi quartics, I (with two parameters)

TL;DR

This paper classifies the 1-Weierstrass points on Kuribayashi quartic curves with two parameters, analyzing their geometric properties and providing a detailed understanding of these special points.

Contribution

It introduces a classification of 1-Weierstrass points on Kuribayashi quartics with two parameters, expanding understanding of their geometric structure.

Findings

Classification of 1-Weierstrass points for the curves

Analysis of the geometric properties of these points

Conditions on parameters for the classification

Abstract

In this paper, we classify the 1-Weierstrass points of the Kuribayashi quartic curves with two parameters $a$ and $b$ defined by the equation \[C_{a,b}:x^{4}+y^{4}+z^{4}+ax^{2}y^{2}+b(x^{2}+y^{2})z^{2}=0,\] such that $(a^2-4)(b^2-4)(a^2-1)(b^{2}-a-2)\neq0.$ Furthermore, the geometry of these points is investigated.