Geometries uniquely embeddable in projective spaces

Abstract

The search for geometric lattices, which admit an embedding in a projective space satisfying universal or uniqueness properties, has been initiated by W.M. Kantor [2,3] (see also the survey [4]). A recent result of the author [8] leads to a unification and generalization of Kantor's theorems (see Theorem 1). A typical application of this generalization is the following (see Section III.D).

Let G be a geometric lattice which is not the union of two hyperplanes. Assume that each interval [p,1] of G, where p is a point, is isometrically embeddable in PG(m-1,K) for some field K, and that all embeddings of [L,1] in PG(m-2,K), where L is a line, are projectively equivalent, then G is isometrically embeddable in PG(m,K) (and its embeddings satisfy some uniqueness properties).