On quartics with the maximal number of the maximal tangency lines

TL;DR

This paper investigates special quartic curves with the maximum number of tangent lines of maximal multiplicity, analyzing their line arrangements, sextactic points, and associated conic configurations.

Contribution

It characterizes Fermat and Komiya-Kuribayashi quartics with maximal tangency lines and studies their sextactic points and conic configurations, revealing new geometric properties.

Findings

Identification of quartics with maximum tangent lines

Analysis of sextactic points on these quartics

Descriptions of related conic configurations

Abstract

In this note, we examine the arrangements of lines and configurations of points that emerge from Fermat (von Dyck) and Komiya-Kuribayashi quartics. These quartics are characterized by having the maximum number of lines of maximal tangency, that is, lines for which the intersection multiplicity at the tangency point is equal to the degree of the curve. Additionally, we delve into the study of sextactic points on these quartics - points at which there exists a conic with the curve having a local intersection multiplicity of at least 6, which is one more than that observed at a general point - alongside the related configurations of conics.