Classification of 1-Weierstrass points on Kuribayashi quartics, II (with three parameters)

TL;DR

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

Contribution

It provides a comprehensive classification of 1-Weierstrass points on Kuribayashi quartics with three parameters, extending previous work and exploring their geometric configurations.

Findings

Explicit classification of 1-Weierstrass points for the curves.

Analysis of the geometric structure of these points.

Conditions ensuring the general position of the points.

Abstract

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