Unterräume von Kettengeometrien und Kettengeometrien mit Quadrikenmodell. (On subspaces of chain geometries and chain geometries with quadrics model)
type: Item
instance of: scholarly article
title: Unterräume von Kettengeometrien und Kettengeometrien mit Quadrikenmodell. (On subspaces of chain geometries and chain geometries with quadrics model)
zbMATH Open document ID: 0733.51003
DOI: 10.1007/BF03323290
author: Hans-Joachim Kroll
published in: Results in Mathematics
publication date: 1991-01-01T00:00:00Z
review text: Any unitary commutative associative K-algebra (A,K) determines a chain geometry \(\Sigma\) (K,A). If (1) (A,K) is cinematic then from \textit{H. Hotje} [J. Geom. 5, 85-94 (1974; Zbl 0283.50020), and Geom. Dedicata 7, 355-362 (1978; Zbl 0395.51007)] there follows (2) the existence of a quadric model of type A for \(\Sigma\) (K,A) which means that the point set of \(\Sigma\) (K,A) can be viewed as a subset of some quadric inside some projective space \(\Pi\) such that chains generate planes of \(\Pi\), and those planes intersect the point set of \(\Sigma\) (K,A) giving back exactly the generating chain. In turn, this implies (3) that \(\Sigma\) (K,A) satisfies the bundle theorem, cf. \textit{M. Werner} [Kettengeometrien und Quadrikenmodelle, (Ph. Thesis, Stuttgart) (1981; Zbl 0474.51005)]. By \textit{W. Benz} [J. Geom. 31, No.1/2, 22-31 (1988; Zbl 0643.51005)] (3) is equivalent to (A,K) being weakly quadratic. One wishes to characterize those algebras which correspond to chain geometries satisfying (2). In this context, the author constructs examples of chain geometries \(\Sigma\) (K,A) which fulfil (2) while (A,K) fails to satisfy (1). His constructions involve subgeometries of a chain geometry and their associated subalgebras which are studied in some detail. Among others he proves: Given a commutative cinematic algebra (A,L), a subfield K of L, and a subalgebra (U,K) of (A,K) such that \(U\cap L=K\), then \(\Sigma\) (K,A) fulfils (2). Now, choosing (U,K) non-cinematic, yields the desired examples.
Mathematics Subject Classification ID: 51B05
Mathematics Subject Classification ID: 16S99
Mathematics Subject Classification ID: 13A99
zbMATH DE Number: 4212891
zbMATH Keywords: quadric model of a chain geometry
zbMATH Keywords: bundle theorem
zbMATH Keywords: cinematic algebra
reviewed by: Horst Szambien
MaRDI profile type: Publication
cites work: Q5674022
cites work: Eine Kennzeichnung des Büschelsatzes in Kettengeometrien. (A characterization of the pencil theorem in chain geometries)
cites work: Einbettung gewisser Kettengeometrien in projektive Räume
cites work: Die Algebren einbettbarer Berührstrukturen
cites work: Q3930314
cites work: Quadrikenmodell einer Kettengeometrie
full work available at URL: https://doi.org/10.1007/bf03323290
OpenAlex ID: W2014325538
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description: scientific article; zbMATH DE number 4212891