Embeddings of Line-grassmannians of Polar Spaces in Grassmann Varieties

An embedding of a point-line geometry Γ is usually defined as an injective mapping ε from the point-set of Γ to the set of points of a projective space such that ε(l) is a projective line for every line l of Γ. However, different situations are considered in the literature, where ε(l) is allowed to be a subline of a projective line or a curve. In this paper we propose a more general definition of embedding which includes all the above situations and we focus on a class of embeddings, which we call Grassmann embeddings, where the points of Γ are firstly associated to lines of a projective geometry PG(V ), next they are mapped onto points of PG(V ∧V ) via the usual projective embedding of the line-grassmannian of PG(V ) in PG(V ∧ V ). In the central part of our paper we study sets of points of PG(V ∧ V ) corresponding to lines of PG(V ) totally singular for a given alternating, hermitian or quadratic form of V . Finally, we apply the results obtained in that part to the investigation of Grassmann embeddings of several generalized quadrangles. MSC 2000: 51A45, 51A50, 51E12, 15A75, 15A69, 14A10.