On Veldkamp lines

Abstract

One says that Veldkamp lines exist for a point-line geometry if, for any three distinct (geometric) hyperplanes A, B and C (i) is not properly contained in B and (ii) A\B C implies C or A B = A\C. Under this condition, the set V of all hyperplanes of acquires the structure of a linear space { the Veldkamp space { with intersections of distinct hyperplanes playing the role of lines. It is shown here that an interesting class of strong parapolar spaces (which includes both the half-spin geometries and the Grassmannians) possess Veldkamp lines. Combined with other results on hyperplanes and embeddings, this implies that for most of these parapolar spaces, the corresponding Veldkamp spaces are projective spaces. The arguments incorporate a model of partial matroids based on intersections of sets.