The Fermat curves, arrangements of lines, and intersections of osculating curves
Torgunn Karoline Moe, Nils Peder Astrup Toft
TL;DR
The paper studies highly symmetric Fermat curves $\mathcal{F}_d$ by combining explicit osculation theory (tangent lines, inflection points, osculating and hyperosculating conics) with group actions, locating $3d^2$ sextactic points on three line grids and constructing new free line arrangements from these grids. It proves that three-grid line configurations $\mathcal{B}$, $\mathcal{M}$, $\mathcal{N}$ yield free curves in several combinations, and it derives invariance properties of osculating-curve intersections under automorphisms that fix lines. A key contribution is a general method to deduce fixed-point intersection patterns for osculating curves of any degree $n$ via automorphisms, plus explicit descriptions of intersection behavior among tangent lines and hyperosculating conics, including when hyperosculating conics share two common points for $d>3$. Overall, the results advance the understanding of free-curve geometry on Fermat curves, provide concrete, symmetry-driven intersection coordinates, and propose a broad, degree-agnostic framework for 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.