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Affine Spaces within Projective Spaces

Andrea Blunck, Hans Havlicek

Abstract

We endow the set of complements of a fixed subspace of a projective space with the structure of an affine space, and show that certain lines of such an affine space are affine reguli or cones over affine reguli. Moreover, we apply our concepts to the problem of describing dual spreads. We do not assume that the projective space is finite-dimensional or pappian.

Affine Spaces within Projective Spaces

Abstract

We endow the set of complements of a fixed subspace of a projective space with the structure of an affine space, and show that certain lines of such an affine space are affine reguli or cones over affine reguli. Moreover, we apply our concepts to the problem of describing dual spreads. We do not assume that the projective space is finite-dimensional or pappian.

Paper Structure

This paper contains 5 sections, 29 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. Affine spaces with point set ${\mathcal{S}}$
  3. The symmetric case
  4. The general case
  5. An application: Dual spreads

Key Result

Lemma 2.1

Theorems & Definitions (31)

  • Lemma 2.1
  • Proposition 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Theorem 2.6
  • Proposition 3.1
  • Remark 3.2
  • Lemma 3.3
  • Corollary 3.4
  • ...and 21 more