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 𝜋 of the lines and each line contains of the points.It is usually denoted by the symbol ,with . A configuration 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 and k in the cubic curve are collinear.