Rank n Geometries with Affine Hyperplanes and Dual Affine Point Residues

Abstract

This paper extends to any rank n a previous classification result [4, 5], on rank 3 geometries with affine planes and dual affine point residues. Let Г be a rank n incidence geometry of points, lines, ..., hyperplanes, the hyperplanes of which are affine spaces, the point residues of which are dual affine spaces and which satisfies Axiom [A]: any two points of Г are incident with at most one line. Such a geometry is necessarily isomorphic to the incidence structure obtained from any n -dimensional affine space A either by deleting a point and all subspaces of A incident with that point or by deleting a direction of lines and all subspaces of A parallel to that direction.