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Pencilled regular parallelisms

Hans Havlicek, Rolf Riesinger

Abstract

Over any field $\mathbb K$, there is a bijection between regular spreads of the projective space ${\rm PG}(3,{\mathbb K})$ and $0$-secant lines of the Klein quadric in ${\rm PG}(5,{\mathbb K})$. Under this bijection, regular parallelisms of ${\rm PG}(3,{\mathbb K})$ correspond to hyperflock determining line sets (hfd line sets) with respect to the Klein quadric. An hfd line set is defined to be \emph{pencilled} if it is composed of pencils of lines. We present a construction of pencilled hfd line sets, which is then shown to determine all such sets. Based on these results, we describe the corresponding regular parallelisms. These are also termed as being \emph{pencilled}. Any Clifford parallelism is regular and pencilled. From this, we derive necessary and sufficient algebraic conditions for the existence of pencilled hfd line sets.

Pencilled regular parallelisms

Abstract

Over any field , there is a bijection between regular spreads of the projective space and -secant lines of the Klein quadric in . Under this bijection, regular parallelisms of correspond to hyperflock determining line sets (hfd line sets) with respect to the Klein quadric. An hfd line set is defined to be \emph{pencilled} if it is composed of pencils of lines. We present a construction of pencilled hfd line sets, which is then shown to determine all such sets. Based on these results, we describe the corresponding regular parallelisms. These are also termed as being \emph{pencilled}. Any Clifford parallelism is regular and pencilled. From this, we derive necessary and sufficient algebraic conditions for the existence of pencilled hfd line sets.

Paper Structure

This paper contains 6 sections, 15 theorems, 26 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main results and examples
  4. Proofs
  5. Back to $\mathop{\mathrm{PG}}\nolimits(3,{\mathbb K})$
  6. Aspects of characteristic two

Key Result

Theorem 3.1

In $\mathop{\mathrm{PG}}\nolimits(5,{\mathbb K})$, let $D$ be a line such that is non-empty. Then, upon choosing any mapping $f\colon D\to{\mathcal{E}}_D$, the union is a pencilled hfd line set.

Figures (1)

  • Figure 1: An hfd line set ${\mathcal{H}}_{12}$

Theorems & Definitions (33)

  • Definition 2.1
  • Theorem 3.1: Construction of pencilled hfd line sets
  • Example 3.2
  • Example 3.3
  • Theorem 3.4: Main theorem on pencilled hfd line sets
  • Remark 3.5
  • Proposition 3.6
  • Remark 3.7
  • Theorem 3.8
  • Remark 3.9
  • ...and 23 more