Sextactic and type-9 points on the Fermat cubic and associated objects

TL;DR

This paper investigates special point configurations on the Fermat cubic, focusing on sextactic and type-9 points, and provides explicit coordinates for all 72 type-9 points, enriching the understanding of their geometric arrangements.

Contribution

It offers explicit coordinate descriptions for all 72 type-9 points on the Fermat cubic and analyzes related line and conic arrangements.

Findings

Explicit coordinates for all 72 type-9 points on the Fermat cubic

Detailed analysis of line and conic arrangements associated with these points

Enhanced understanding of geometric configurations on the Fermat cubic

Abstract

In this note we study line and conic arrangements associated to sextactic and type 9 points on the Fermat cubic $F$ and we provide explicit coordinates for each of the 72 type 9 points on $F$.