Finite rank 3 geometries with affine planes and dual affine point residues

Abstract

Let Γ be a rank 3 incidence geometry of points, lines and planes. This paper classifies all finite geometries Γ whose planes are affine, whose point residues are dual affine and which satisfy Condition 1: any two points of Γ are incident with at most one line. Such a geometry is necessarily isomorphic to the incidence structure obtained from any 3-dimensional affine space either by deleting a point and all lines and planes through that point or by deleting a direction of lines and all planes parallel to that direction.