The Fermat curves, arrangements of lines, and intersections of osculating curves

TL;DR

This paper investigates the geometric arrangements of lines and osculating curves related to Fermat curves, revealing new grid structures, free curves, and intersection properties influenced by automorphism groups.

Contribution

It introduces new line arrangements from sextactic points, analyzes hyperosculating conics, and generalizes intersection results for osculating curves of any degree.

Findings

Sextactic points on Fermat curves form three grid patterns.

Hyperosculating conics intersect in a structured manner.

Automorphism groups influence osculating curve intersections.

Abstract

In this paper we present new results about arrangements of lines and osculating curves associated to the Fermat curves in the projective plane. We first consider the sextactic points on the Fermat curves and show that they are distributed on three grids. The grid lines constitute new line arrangements and examples of free curves associated with the Fermat curves. Moreover, we compute the hyperosculating conics to the Fermat curves, study the arrangement of these conics, and find that they intersect in a special way. The latter result is a consequence of the action of the group of automorphisms on osculating curves, and we conclude with a more general result for intersections of osculating curves of any given degree.