Embeddings of Line-grassmannians of Polar Spaces in Grassmann Varieties

An embedding of a point-line geometry \Gamma is usually defined as an injective mapping \epsilon from the point-set of \Gamma to the set of points of a projective space such that \epsilon(l) is a projective line for every line l of \Gamma, but different situations have lately been considered in the literature, where \epsilon(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 Grassmman embeddings, where the points of \Gamma are firstly associated to lines of a projective geometry PG(V), next they are mapped onto points of PG(V\wedge V) via the usual projective embedding of the line-grassmannian of PG(V) in PG(V\wedge V). In the central part of our paper we study sets of points of PG(V\wedge V) corresponding to lines of PG(V) totally singular for a given pseudoquadratic form of V. Finally, we apply the results obtained in that part to the investigation of Grassmann embeddings of several generalized quadrangles

Lax embeddings of the Hermitian Unital

In this paper, we prove that every lax generalized Veronesean embedding of the Hermitian unital U of PG(2, L), L a quadratic extension of the field K and |K| ≥ 3, in a PG(d, F), with F any field and d ≥ 7, such that disjoint blocks span disjoint subspaces, is the standard Veronesean embedding in a subgeometry PG(7, K ) of PG(7, F) (and d = 7) or it consists of the projection from a point p ∈ U of U \ {p} from a subgeometry PG(7, K ) of PG(7, F) into a hyperplane PG(6, K ). In order to do so, when |K| > 3 we strongly use the linear representation of the affine part of U (the line at infinity being secant) as the affine part of the generalized quadrangle Q(4, K) (the solid at infinity being non-singular); when |K| = 3, we use the connection of U with the generalized hexagon of order 2

Grassmann embeddings of polar Grassmannians

In this paper we compute the dimension of the Grassmann embeddings of the polar Grassmannians associated to a possibly degenerate Hermitian, alternating or quadratic form with possibly non-maximal Witt index. Moreover, in the characteristic $2$ case, when the form is quadratic and non-degenerate with bilinearization of minimal Witt index, we define a generalization of the so-called Weyl embedding (see [I. Cardinali and A. Pasini, Grassmann and Weyl embeddings of orthogonal Grassmannians. J. Algebr. Combin. 38 (2013), 863-888]) and prove that the Grassmann embedding is a quotient of this generalized "Weyl-like" embedding. We also estimate the dimension of the latter.Comment: 25 pages/revised version after revie

Generalized Veronesean embeddings of projective spaces, part II: the lax case

We classify all embeddings theta : PG(n,K) -> PG(d, F), with d >= n(n+3)/2 and K, F skew fields with vertical bar K vertical bar > 2, such that 0 maps the set of points of each line of PG(n,K) to a set of coplanar points of PG(d, F), and such that the image of theta generates PG(d, F). It turns out that d = 1/2n(n + 3) and all examples "essentially" arise from a similar "full" embedding theta' : PG(n, K) -> PG(d,K) by identifying K with subfields of IF and embedding PG(d, K) into PG(d, F) by several ordinary field extensions. These "full" embeddings satisfy one more property and are classified in [5]. They relate to the quadric Veronesean of PG(n, K) in PG(d, K) and its projections from subspaces of PG(d, K) generated by sub-Veroneseans (the point sets corresponding to subspaces of PG(n,K)), if K is commutative, and to a degenerate analogue of this, if K is noncommutative