Ferdinand Douwe Veldkamp: 5 September 1931–3 August 1999

On 3 August 1999, Ferdinand Douwe Veldkamp, one of the managing editors of Geometriae Dedicata since 1981 passed away. Professor Veldkamp was a passionate mathematician. His area of research comprised large parts of algebra and geometry; it extended from algebraic groups and Lie algebras to ring geometries and their axiomatizations. Veldkamp's pioneering work on polar spaces has become a lasting part of the theory of buildings. From octonian geometries, especially Moufang Hjelmslev planes, as a starting point he opened completely new vistas into the geometry of rings. This becomes evident if one observes that the most general ring geometries which can be reasonably treated are today called Veldkamp planes.     

Extended dual affine planes

We study a class of diagram geometries, achieve a characterization of extended dual affine planes, and embed extended dual affine planes in extended projective planes. The geometries studied are rank 3 diagram geometries such that the residue of a point is a dual net, and the residue of a plane is linear; the dual of such a geometry has partitions on lines and planes which are reminiscent of parallelism of lines and planes of an affine 3-space. Examples of these geometries (some in dual form) include extended dual affine planes, Laguerre planes, 3-nets, and orthogonal arrays of strength 3. Theorem: Any such finite geometry satisfying Buekenhout's intersection property, and such that any two points are coplanar, is an extended dual affine plane (and has order 2, 4, or 10). Theorem: This geometry may be embedded in an extended projective plane of the same order.This research was partially supported by NSF Grant MCS-8102361.     

Homomorphisms of Projective Remoteness Planes

A characterization of a class of homomorphisms of projective remoteness planes in terms of their coordinate rings is given. A remoteness preserving homomorphism of projective remoteness planes is factored into three homomorphisms of known types. Two of these are constructed from groups associated with the planes and the homomorphism that induces on the coordinate rings of the planes. The third is a covering of planes coordinatized by the same ring. This generalizes known results for projective planes, projective ring planes, and Moufang–Veldkamp planes.     

Neighbor-preserving epimorphisms between projective Klingenberg planes

This paper deals with neighbor-preserving epimorphisms between arbitrary projective Klingenberg planes. Our main result is an algebraic characterization of such epimorphisms which generalizes a theorem of J. C. Ferrar and F. D. Veldkamp [2] for neighbor-preserving epimorphisms between projective ring planes     

Distant-preserving epimorphisms between projective Klingenberg planes

This paper deals with epimorphisms between arbitrary projective Klingenberg planes which preserve the non-neighbor relation. Our main result is an algebraic characterization of such epimorphisms which generalizes a theorem of F. D. Veldkamp [7] for distant preserving epimorphisms between projective ring planes.     

On Barbilian spaces in projective lattice geometries

We introduce the notion of a Barbilian space of a projective lattice geometry in order to investigate the relationship between lattice-geometric properties and the properties of point-hyperplane structures associated with. We obtain a characterization of those projective lattice geometries, the Barbilian space of which is a Veldkamp space.     

Characterizing Combinatorial Geometries by Numerical Invariants

We show that the projective geometry PG(r − 1,q ) for r & 3 is the only rank- r(combinatorial) geometry with (q^r − 1) / (q − 1) points in which all lines have at least q + 1 points. For r = 3, these numerical invariants do not distinguish between projective planes of the same order, but they do distinguish projective planes from other rank-3 geometries. We give similar characterizations of affine geometries. In the core of the paper, we investigate the extent to which partition lattices and, more generally, Dowling lattices are characterized by similar information about their flats of small rank. We apply our results to characterizations of affine geometries, partition lattices, and Dowling lattices by Tutte polynomials, and to matroid reconstruction. In particular, we show that any matroid with the same Tutte polynomial as a Dowling lattice is a Dowling lattice.     

Partial circle geometries and antiregular quadrangles

A new and rather general definition of circle geometries is given. This definition is such that circle planes and chain spaces are circle geometries. Also the geometry of points and traces of an antiregular quadrangle is a partial circle geometry. Orthogonal quadrangles can then be characterised as those antiregular generalised quadrangles where in the associated partial circle geometry the Miquel condition is satisfied.     

Geometries of type [L.Af*]

Let be a [L.Af*]-geometry, that is a rank 3 geometry with linear spaces as plane residues, with dual affine planes as point residues and with generalized digons as line residues. Assume that (LL) holds in . In the particular case where the plane residues are finite circles, the structure of such geometries has been strongly restricted by A. P. Sprague. Moreover, C. Lefèvre and L. Van Nypelseer have given a complete classification of such geometries under the assumption that the plane residues are affine planes. We generalize these two results for [L.Af*]-geometries.Aspirant du Fonds National Belge de la Recherche Scientifique.     

Finite geometries which contain dual affine planes

Finite geometries in which each plane is projective or dual affine over the field of two elements, or affine over the field of three elements, are studied. It is shown that no connected geometry can mix all three species of planes, and the geometries in which projective and dual affine planes occur are classified.