Sextactic points on the Fermat cubic curve and arrangements of conics

TL;DR

This paper explores the geometry of 6-division points on the Fermat cubic and related conics, using symbolic algebra to suggest new research directions for understanding these configurations and their generalizations.

Contribution

It reports on the geometric properties of division points on the Fermat cubic and proposes new research avenues for studying related algebraic curves and configurations.

Findings

Identifies geometric configurations of 6-division points on the Fermat cubic.

Uses symbolic algebra to derive properties of associated conics.

Proposes multiple directions for future research in algebraic geometry.

Abstract

The purpose of this note is to report, in narrative rather than rigorous style, about the nice geometry of $6$-division points on the Fermat cubic $F$ and various conics naturally attached to them. Most facts presented here were derived by symbolic algebra programs and the idea of the note is to propose a research direction for searching for conceptual proofs of facts stated here and their generalisations. Extensions in several directions seem possible (taking curves of higher degree and contact to $F$, studying higher degree curves passing through higher order division points on $F$, studying curves passing through intersection points of already constructed curves, taking the duals etc.) and we hope some younger colleagues might find pleasure in following proposed paths as well as finding their own.