Polarities in planar spaces
type: Item
instance of: scholarly article
title: Polarities in planar spaces
zbMATH Open document ID: 1033.51003
author: Paola Biondi
published in: Journal of Geometry
publication date: 2002-08-05T00:00:00Z
review text: A linear space is a pair \((\mathcal S,\mathcal L)\), where \(\mathcal S\) is a set of points and \(\mathcal L\) is a set of subsets of \(\mathcal S\), called lines, such that any line contains at least two points and any two distinct points lie on exactly one line. A subspace of a linear space is a set of points containing the line through any two of its distinct points. A planar space is a triple \((\mathcal S,\mathcal L,\mathcal P)\), where \((\mathcal S,\mathcal L)\) is a linear space and \(\mathcal P\) is a set of proper subspaces of \((\mathcal S,\mathcal L)\), called planes, such that through every three non-collinear points there is exactly one plane and every plane contains at least three non-collinear points. A planar space is locally projective if for every pair of distinct planes their intersection is either a line or the empty set. Let \({\mathbb P}_3\) be a \(3\)-dimensional projective space and let \(\mathcal S\) be a subset of the point-set of \({\mathbb P}_3\). If no line of \({\mathbb P}_3\) meets \(\mathcal S\) in exactly one point, then \((\mathcal S,\mathcal L,\mathcal P)\), where \(\mathcal L=\{\ell \cap \mathcal S: \ell\) line of \({\mathbb P}_3 \}\) and \(\mathcal P=\{\pi \cap \mathcal S: \pi \) plane of \({\mathbb P}_3\}\) is a locally projective planar space which is said to be embedded in \({\mathbb P}_3.\) In the last thirty years many authors tried to work on the following problem: Is any locally projective planar space embedded in a \(3\)-dimensional projective space? The problem seems to be too difficult without further conditions on the planar space. All results on this problem have been obtained putting some additional arithmetical or geometrical conditions on the locally projective planar space. In the paper under review the author considers the previous problem assuming that the locally projective planar space is equipped with an involutorial permutation of the line-set. Let \((\mathcal S,\mathcal L,\mathcal P)\) be a locally proje…
reviewed by: Nicola Durante
Mathematics Subject Classification ID: 51A45
zbMATH DE Number: 1760620
zbMATH Keywords: linear spaces
zbMATH Keywords: embedding into 3-dimensional projective space
zbMATH Keywords: polarities
zbMATH Keywords: locally projective planar space
MaRDI profile type: Publication
full work available at URL: s00022-001-8564-2
OpenAlex ID: W1978273868
DOI: 10.1007/S00022-001-8564-2
description: scientific article; zbMATH DE number 1760620