Geradenmodelle der M�bius- und Burau-Geometrien

Abstract

Let (L, K) be an arbitrary (not necessary abelian) field extension, let V be a (left) vector space over L (and therefore over the subfield K) and let μ be the set of all sub-spaces of the projective space II(V, K). Then the canonical map ϰ: P ≔ V*/K* → U given by ϰ: K*a → L*a/K* ≔ {K*(λa)∣λ ∈ L*{ induces a spreadS in Π(V, K). Furthermore ϰ induces a chain structure by applying ϰ on certain subspaces of U. Our purpose is to give a synthetic description of these chain structures. For a regular SpreadS, i.e. K is commutative, the chain structure is a generalization of the Burau geometries [11]. In this paper we represent the Staudt chains by Segre manifolds.