Incidence and Combinatorial Properties of Linear Complexes

TL;DR

This paper introduces a generalized notion of polarity to describe linear complexes of subspaces in incidence geometry, revealing new combinatorial structures and properties of these complexes.

Contribution

It defines a generalized polarity framework that characterizes linear complexes and explores their combinatorial and incidence properties, including the structure of line partitions.

Findings

Generalized polarity links to linear complexes with null property.

Existence of linear complexes without stars remains an open problem.

Characterization of line partitions with linearity property.

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.