An arithmetic reduction of finite rank 3 geometries with linear spaces as plane residues and with dual linear spaces as point residues

Abstract

Let Г be a rank three incidence geometry of points, lines and planes whose planes are linear spaces and whose point residues are dual linear spaces (notice that we do not require anything on the line residues). We assume that the residual linear spaces of Г belong to a natural class of finite linear spaces, namely those linear spaces whose full automorphism group acts flag-transitively and whose orders are polynomial functions of some prime number. This class consists of six families of linear spaces. In Г the amalgamation of two such linear spaces imposes an equality on their orders leading, in particular, to a series of diophantine equations, the solutions of which provide a reduction theorem on the possible amalgams of linear spaces that can occur in Г.We prove that one of the following holds (up to a permutation of the words “point” and “plane”): A) the planes of Г and the dual of the point residues belong to the same family and have the same orders B) the diagram of Г is in one of six families C) the diagram of Г belongs to a list of seven sporadic cases. Finally, we consider the particular case where the line residues of Г are generalized digons.