Designs embeddable in a plane cubic curve (Part 2 of Planar projective configurations)

Abstract

A configuration or a design K is a system of p points and m lines such that each point lies on ? of the lines and each line contains $\mu$ of the points.It is usually denoted by the symbol $(p_?, m_\mu$,with $p?=m\mu$. A configuration $K= (p_?, m_\mu)$ is said to have a geometric representation if we can draw it in the given geometry meaning that the points and lines of K correspond to points and lines in the geometry such that a point is incident with a line in K iff the same is true in the corresponding geometry. In this paper, we consider the problem of representing such combinatorial designs in the geometry of non-singular cubic curves over the complex projective plane. i. e. we study the problem of embedding them into a non-singular cubic curve in the complex projective plane in such a way that (ijk) is an element of the combinatorial design iff the points corresponding to $i,j$ and k in the cubic curve are collinear.