Unterräume von Kettengeometrien und Kettengeometrien mit Quadrikenmodell. (On subspaces of chain geometries and chain geometries with quadrics model)
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.
scientific article; zbMATH DE number 4212891
scientific article