Partial lineations between arguesian projective spaces
type: Item
instance of: scholarly article
title: Partial lineations between arguesian projective spaces
zbMATH Open document ID: 1019.51001
published in: Archiv der Mathematik
publication date: 2002-12-01T00:00:00Z
review text: Over the last years the author and \textit{A. Fröhlicher} developed a new approach to the fundamental theorems of projective geometry, cf. their joint book [Modern projective geometry (Dordrecht) (2000; Zbl 0972.51001)]. In the present paper the author extends this approach to partial lineations between arguesian projective spaces. A partial lineation between projective spaces \(P\) and \(Q\) is a mapping \(g:P\setminus E \to Q\) defined outside a subset \(E \subset P\) which preserves collinearity and which maps every line that intersects \(E\) to a single point. The mapping \(g\) is called thick if there exists no line in \(P\) whose image consists of precisely two points, and \(g\) is called nondegenerate if it is thick and its image is not contained in a line. The author proves that every nondegenerate partial lineation between arguesian projective spaces is induced by a generalized semilinear map of the underlying vector spaces. Here, a generalized semilinear map is a map satisfying all the usual requirement of a semilinear map where a place is used instead of a field isomorphism. This theorem extends and sharpens several known results scattered in the literature.
reviewed by: Norbert Knarr
Mathematics Subject Classification ID: 51A10
zbMATH DE Number: 1837316
zbMATH Keywords: fundamental theorems of projective geometry
zbMATH Keywords: partial lineations between arguesian projective spaces
zbMATH Keywords: generalized semilinear map
author: C. Faure
MaRDI profile type: Publication
full work available at URL: s00013-002-8317-y
OpenAlex ID: W2071533878
DOI: 10.1007/S00013-002-8317-Y
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description: scientific article; zbMATH DE number 1837316