Baer subspaces within Segre manifolds

Abstract

Baer subspaces of projective spaces as well as Segre manifolds of pappian projective spaces are very well known. But seemingly they are unrelated topics, apart from the (more or less formal) fact that both of them may be described in terms of tensor products of vector spaces. Baer subspaces of a desarguesian projective space with an underlying (not necessarily commutative) field L arise from subfields K of right and left degree 2 over L. (Recall that the right and left degree of a field extension may be different; cf. [4,123ff).) If m is a right vector space over K, then the tensor product m®KL is a right vector space over L. With 1 E L we have the canonical embedding 1)) ~ 1))®1 of m in m ®K L. This yields an embedding of the projective space on m in the projective space on m ®K L as a Baer subspace. When n1 and nz are vector spaces over the same commutative field L, then the set of all non-zero pure bivectors of n1 ®L nz determines a Segre manifold in the projective space on n1 ®LnZ. Following geometric ideas in [3] and [10], a definition of Segre manifolds will be given when the ground field L is arbitrary. However, by following this definition, the connection to tensor products of vector spaces seems to be lost when L is a skew field, since forming n1 ®L Ilz requires a rig h t vector space III and ale f t vector space Ilz over L. And this is not in accordance with the geometric approach. The following construction of a Baer subspace within a Segre manifold essentially depends on the existence of an element a E L which has degree two over the centre of