Af∗ . Af geometries, the Klein quadric and [formula omitted]nq

Abstract

An Af ∗. Af geometry of order q is a residually connected rank three geometry where planes are dual affine planes and stars of points are affine planes of order q. We prove that such a geometry is necessarily obtained from the Klein quadric Q + 5( q) of PG(5, q) deleting the points of a hyperplane and considering as points the elements of one of the two systems of maximal subspaces of Q= Q + 5( q), as lines the points of Q, and as planes the elements of the other system. The deleted hyperplane is tangent to Q if and only if the Af ∗. Af geometry obtained satisfies property (PL 1) (i.e. there is a unique plane on every point-line antiflag). When (PL 1) is satisfied, some generalizations are obtained for L ∗ . L (resp. N ∗. L) geometries (i.e. residually connected rank three geometries where planes are dual linear spaces (resp. dual nets) and stars of points are linear spaces). In particular, this yields a characterization of H n q and T ∗ 2( K), where K = AG(2, q), in the context of rank 3 partial geometries. Furthermore, it leads to some classification results for other rank n⩾3 diagrams related to what we call the rank n H n q ( n⩾3, see Examples 5.1 and 5.4 for the definition).