On the configurations of nine pointson a cubic curve

We study the reciprocal position of nine points in the plane, accordingto their collinearities. In particular, we consider the case in which thenine points are contained in an irreducible cubic curve and we give theirclassification. If we call two configurations different when the associatedincidence structures are not isomorphic, we see that there are 131 config-urations that can be realized inP2Q, and there are two more inP2K, whereK=Q[√−3] (one of the two is the Hesse configuration given by thenine inflection points of a cubic curve). Finally, we compute the possibleHilbert functions of the ideals of the nine points.