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A note on Clifford parallelisms in characteristic two

Hans Havlicek

Abstract

It is well known that a purely inseparable field extension $L/F$ with some extra property and degree $[L:F]=4$ determines a Clifford parallelism on the set of lines of the three-dimensional projective space over $F$. By extending the ground field of this space from $F$ to $L$, we establish the following geometric description of such a parallelism in terms of a distinguished `absolute pencil of lines' of the extended space: Two lines are Clifford parallel if, and only if, there exists a line of the absolute pencil that meets both of them.

A note on Clifford parallelisms in characteristic two

Abstract

It is well known that a purely inseparable field extension with some extra property and degree determines a Clifford parallelism on the set of lines of the three-dimensional projective space over . By extending the ground field of this space from to , we establish the following geometric description of such a parallelism in terms of a distinguished `absolute pencil of lines' of the extended space: Two lines are Clifford parallel if, and only if, there exists a line of the absolute pencil that meets both of them.

Paper Structure

This paper contains 4 sections, 7 theorems, 26 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Algebraic preliminaries
  3. The absolute pencil
  4. Future Research

Key Result

Lemma 2.1

The L-algebra $L_{(L)}$ is local and quadratic. The ideal of non-invertible elements of $L_{(L)}$ is the kernel $\Pi$ of the homomorphism $\pi$ from eq:pi.

Figures (1)

  • Figure 1: Sublines in the absolute plane $\Pi$.

Theorems & Definitions (20)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Remark 3.3
  • Definition 3.4
  • Proposition 3.5
  • ...and 10 more