On the configurations of nine points on a cubic curve

TL;DR

This paper classifies all possible configurations of nine points on an irreducible cubic curve in the plane, identifying 131 realizable configurations over rationals and additional ones over a quadratic extension, and analyzes their algebraic properties.

Contribution

It provides a complete classification of nine-point configurations on cubic curves, including their incidence structures and Hilbert functions, expanding understanding of geometric and algebraic relations.

Findings

131 configurations over ield

2 additional configurations over quadratic extension

Computed Hilbert functions of the point ideals

Abstract

We study the reciprocal position of nine points in the plane, according to their collinearities. In particular, we consider the case in which the nine points are contained in an irreducible cubic curve and we give their classification. If we consider two configurations different when the associated incidence structures are not isomorphic, we see that there are 131 configurations that can be realized in $\mathbb{P}^2_{\mathbb{Q}}$, and there are two more in $\mathbb{P}^2_K$, where $K =\mathbb{Q}[\sqrt{-3}]$ (one of the two is the Hesse configuration given by the nine inflection points of a cubic curve). Finally, we compute the possible Hilbert functions of the ideals of the nine points.