Einbettung gewisser Kettengeometrien in projektive Räume

Abstract

The theory of “chain geometries” as represented in [2] is a generalisation of the concept of Mobius-, Laguerre- and pseudo-euclidean planes over a commutative field K. It is well known that these geometries can be represented as a 2-dimensional variety of the 3-dimensional projective space over K. It will be shown how to embed in a similar way a class of “chain geometries”, which covers these planes. The algebras belonging to these geometries are the kinematic algebras, studied by H.KARZEL, in which x2∃ Kx+K for each element x of the algebra. If the algebra is of rank n the geometry will be represented on a n-dimensional algebraic variety of the (n+1)-dimensional projective space π, the chains being the intersection of with planes of π having no line but at least two points in common with .