Incidence and Combinatorial Properties of Linear Complexes

Abstract

In this paper a generalisation of the notion of polarity is exhibited which allows to completely describe, in an incidence-geometric way, the linear complexes of $h$-subspaces. A generalised polarity is defined to be a partial map which maps $(h-1)$-subspaces to hyperplanes, satisfying suitable linearity and reciprocity properties. Generalised polarities with the null property give rise to a linear complexes and vice versa. Given that there exists for $h>1$ a linear complex of $h$-subspaces which contains no star --this seems to be an open problem over an arbitrary ground field --the combinatorial structure of a partition of the line set of the projective space into non-geometric spreads of its hyperplanes can be obtained. This line partition has an additional linearity property which turns out to be characteristic.