Geometric hyperplanes of the half-spin geometries arise from embeddings

Abstract

Let the point-line geometry = ( P;L) be a half-spin geometry of type Dn;n. Then, for every embedding of in the projective space P(V ), where V is a vector space of dimension 2 n 1 , it is true that every hyperplane of arises from that embedding. It follows that any embedding of this dimension is universal. There are no embeddings of higher dimension. A corollary of this result and the fact that Veldkamp lines exist ([6]), is that the Veldkamp space of any half-spin geometry (n 4) is a projective space.