Some Characterizations of Finite 3-Dimensional Projective Spaces and Affino-Projective Planes

Abstract

This chapter presents some characterizations of finite 3-dimensional projective spaces and affino­projective planes. A linear space is a non-empty set of elements called points together with a family of distinguished subsets called lines such that any two distinct points are contained in exactly one line, each line containing at least two points. A linear space is said finite if it has a finite number of points. A transversal of two lines L and L′ will be any line intersecting L ∪ L′ in two distinct points. The chapter explains the theorem where S is considered as a finite linear space. If there is a non-negative integer t such that for any two disjoint lines L, L′ of S, any point p outside L and L′ is on exactly t transversals of L and L′, then one of the following possibilities occurs: (1) S is a generalized projective space, (2) S is an affine plane, an affine plane with one point at infinity, or a punctured projective plane, and (3) S is the Fano quasi-plane.