Subplane covered nets and semipartial geometries
type: Item
instance of: scholarly article
title: Subplane covered nets and semipartial geometries
zbMATH Open document ID: 0760.51011
DOI: 10.1016/0012-365X(92)90539-R
published in: Discrete Mathematics
publication date: 1993-01-16T00:00:00Z
review text: Let \({\mathbf N}\) be a net and assume the existence of a set \({\mathbf B}\) of affine subplanes of \({\mathbf N}\) satisfying the following requirements: (i) Each plane in \({\mathbf B}\) has the same set of parallel classes as \({\mathbf N}\); (ii) for each distinct pair of collinear points \(p\), \(p'\) of \({\mathbf N}\), there exists a unique plane in \({\mathbf B}\) containing \(p\) and \(p'\). Then \({\mathbf N}\) is called a subplane covered net. (In the special case, where each plane in \({\mathbf B}\) is a Baer subplane, \({\mathbf N}\) is a derivable net. Hence subplane-covered nets generalize the well-known derivable nets.) The authors completely characterize the finite subplane- covered nets as those nets which are determined by an \((n-1)\)-regulus in \(PG(2n-1,q)\). This extends a previous result by the second author [Abh. Math. Semin. Univ. Hamb. 58, 245-253 (1988; Zbl 0686.51002); Part II: Arch. Math. 55, No. 1, 94-104 (1990; Zbl 0721.51004)], where the finite derivable nets were characterized as those nets determined by a regulus in \(PG(3,q)\).
reviewed by: Dieter Jungnickel
Mathematics Subject Classification ID: 51E14
zbMATH DE Number: 94517
zbMATH Keywords: net
zbMATH Keywords: semipartial geometry
zbMATH Keywords: derivable net
author: F. De Clerck
author: Norman L. Johnson
MaRDI profile type: Publication
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full work available at URL: 0012-365x(92)90539-r
OpenAlex ID: W2071764853
description: scientific article; zbMATH DE number 94517