Publications

A point-line space is an abstract geometric object that consists of a set of points and a set of lines such that on each line there are at least two points. A large class of point-line spaces with high symmetry comes along with buildings, combinatorial objects that are introduced by Jacques Tits and help to study algebraic objects with geometric methods.To formulate quantum mechanics as abstract and general as possible, the physicist Pascual ...

Parapolar spaces are point-line geometries introduced as a geometric approach to (exceptional) algebraic groups. We characterize a wide class of Lie geometries as parapolar spaces satisfying a simple intersection property. In particular, many of the exceptional Lie incidence geometries occur. In an appendix, we extend our result to the locally disconnected case and discuss the locally disconnected case of some other well-known characterizations.

In this paper, we consider a set L of lines of PG(5, q) with the properties that (1) every plane contains 0, 1 or q + 1 elements of L, (2) every solid contains no more than q(2) + q + 1 and no less than q + 1 elements of L, and (3) every point of PG(5, q) is on q + 1 members of L, and we show that, whenever (4) q not equal 2 (respectively, q = 2) and the lines of L through some point are contained in a solid (respectively, a plane), then L is ...

We introduce a rank 3 geometry for any Ree group over a not necessarily perfect field and show that its full collineation group is the automorphism group of the corresponding Ree group. A similar result holds for two rank 2 geometries obtained as a truncation of this rank 3 geometry. As an application, we show that a polarity in any Moufang generalized hexagon is unambiguously determined by its set of absolute points, or equivalently, its set of ...

In 1954 Segre proved the following celebrated theorem : In PG(2, q), with q odd, every oval is a nonsingular conic. Crucial for the proof is Segre's Lemma of Tangents, where a strong result is deduced from the simple fact that the product of the nonzero elements of GF(q) is -1. Relying on this Lemma of Tangents he was able to prove excellent theorems on certain point sets in PG(2,q). To this end he also generalized the classical theorem of ...

Hjelmslev-Moufang planes are point-line geometries related to the exceptional algebraic groups of type E6. More generally, point-line geometries related to spherical Tits-buildingsU+2014Lie incidence geometriesU+2014are the prominent examples of parapolar spaces: axiomatically defined geometries consisting of points, lines and symplecta (structures isomorphic to polar spaces). In this paper we classify the parapolar spaces with a similar ...

We generalize some known results regarding hyperplanes and projective embeddings of point-line geometries with three points per line to geometries with an arbitrary but finite number of points on each line. In order to generalize these results, we need to introduce the new notions of pseudo-hyperplane, (universal) pseudo-embedding, pseudo-embedding rank and pseudo-generating rank. We prove several connections between these notions and address ...

A hyperoval of a point-line geometry is a nonempty set of points meeting each line in either 0 or 2 points. We discuss a combination of theoretical and practical techniques that are helpful for classifying hyperovals of generalized quadrangles. These techniques are based on the connection between hyperovals, even sets and pseudo-embeddings of point-line geometries.