On the configurations of nine points on a cubic curve

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.