Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Chow's Theorem for Linear Spaces

Hans Havlicek

TL;DR

The second possibility can only occur when ( P, L ) and ( P ′, L ′) are 3-dimensional generalized projective spaces.

Abstract

If $φ: L\to L'$ is a bijection from the set of lines of a linear space $(P,L)$ onto the set of lines of a linear space $(P',L')$ ($\dim P, \dim P'\geq 3$), such that intersecting lines go over to intersecting lines in both directions, then $φ$ is arising from a collineation of $(P,L)$ onto $(P',L')$ or a collineation of $(P,L)$ onto the dual linear space of $(P',L')$. However, the second possibility can only occur when $(P,L)$ and $(P',L')$ are 3-dimensional generalized projective spaces.

Chow's Theorem for Linear Spaces

TL;DR

The second possibility can only occur when ( P, L ) and ( P ′, L ′) are 3-dimensional generalized projective spaces.

Abstract

If is a bijection from the set of lines of a linear space onto the set of lines of a linear space (), such that intersecting lines go over to intersecting lines in both directions, then is arising from a collineation of onto or a collineation of onto the dual linear space of . However, the second possibility can only occur when and are 3-dimensional generalized projective spaces.

Paper Structure

This paper contains 4 sections, 2 theorems, 10 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Line Geometry
  4. Chow's Theorem

Key Result

Lemma 1

Let $\dim\hbox{$({\cal P},{\cal L})$}\geq 2$ and let ${\cal M}\subset{\cal L}$ be a maximal related set different from a star of lines. Then the subspace ${\cal E}$ spanned by the lines of ${\cal M}$ is a plane satisfying

Theorems & Definitions (2)

  • Lemma 1
  • Theorem 1