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Incidence and Combinatorial Properties of Linear Complexes

Hans Havlicek, Corrado Zanella

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.

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 -subspaces. A generalised polarity is defined to be a partial map which maps -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 a linear complex of -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.

Paper Structure

This paper contains 7 sections, 20 theorems, 19 equations.

Table of Contents

  1. Introduction
  2. The Grassmannians of a projective space
  3. Linear mappings
  4. Primes and linear complexes
  5. Generalised polarities arising from a linear complex
  6. Existence of singular $(h-1)$-subspaces
  7. Linear line partitions

Key Result

Theorem 1

Br73Ha81 Let $\Sigma$ and $\Sigma'$ be projective spaces, and let $\chi: \Sigma\rightarrow\Sigma'$ be a linear mapping. Then the partial map $\chi$ splits into a projection from $\mathbb A(\chi)$ onto a complementary subspace in $\Sigma$, say $U$, and a collineation between $U$ and a subspace of $\S

Theorems & Definitions (34)

  • Theorem 1
  • Theorem 2
  • Proposition 3
  • proof
  • Theorem 4
  • proof
  • Proposition 5
  • proof
  • Theorem 6
  • proof
  • ...and 24 more