The structure of full polarized embeddings of symplectic and Hermitian dual polar spaces

Abstract

Let ∆ be a thick dual polar space of rank n ≥ 2 and let e be a full polarized embedding of ∆ into a projective space Σ. For every point x of ∆ and every i ∈ {0, . . . , n}, let Ti(x) denote the subspace of Σ generated by all points e(y) with d(x, y) ≤ i. We show that Ti(x) does not contain points e(z) with d(x, z) ≥ i + 1. We also show that there exists a well-defined map e x i from the set of (i − 1)-dimensional subspaces of the residue Res∆(x) of ∆ at the point x (which is a projective space of dimension n − 1) to the set of points of the quotient space Ti(x)/Ti−1(x). In this paper we study the structure of the maps e x i and the subspaces Ti(x) for some particular full polarized embeddings of the symplectic and the Hermitian dual polar spaces. Our investigations allow us to answer some questions asked in the literature.

Polarized and homogeneous embeddings of dual polar spaces

“…Following Cardinali and De Bruyn [9], for every point x ∈ P we define T −1 (x) := 0 (vector notation) and, for every i = 0, 1, ..., n we put T i (x) := e(Γ * i (x)) (regarded as a subspace of V ). In particular, T n (x) = e(Γ * n (x)) = V .…”

Hyperplanes of Hermitian dual polar spaces of rank 3 containing a quad

“…Let V x and V y be the 10-dimensional subspaces of V * such that Σ x = PG(V x ) and Σ y = PG(V y ). Then V * = V x ⊕ V y by [2,Corollary 3.3]. Let α be the plane of H(5, F ) corresponding to y and let U be the 3-space of V 6 corresponding to α.…”

“…, and if σ denotes the unique member of Π containing π Q 1 (w), then (2) and (3) imply that H Z ∩ Q 3 = π Q 3 (σ) = H * Z ∩ Q 3 . Since we now know that H Z = H * Z , the hyperplane H * Z must arise from the embedding Z .…”

Hyperplanes of $DW(5,{\mathbb{K}})$ with ${\mathbb{K}}$ a perfect field of characteristic 2

“…[12]. By [6,Theorem 1.3], for every hyperplane α 2 of α 2 through e(z i ), the set of lines L through z i for which e(L) ⊆ α 2 is a conic C(α 2 ) of Res(z i ). Moreover, there exist reference systems in Res(z i ) and the quotient space α 2 /e(z i ) such that if α 2 /e(z i ) is given by the equation…”

“…In case (3), D H consists of all quads which contain a line of the grid which defines H. In case (4), D H consists of the unique quad which carries the ovoid which defines H. In case (5), D H defines a nonempty and nondegenerate conic in the dual projective plane of Res(π Q (x)). In case (6), D H = ∅ and there exists a quad Q for which Q ∩ H is not a singular hyperplane of Q.…”