On the simple connectedness of hyperplane complements in dual polar spaces

Let Δ be a dual polar space of rank n≥4, H be a hyperplane of Δ and Γ?Δ?H be the complement of H in Δ. We shall prove that, if all lines of Δ have more than 3 points, then Γ is simply connected. Then we show how this theorem can be exploited to prove that certain families of hyperplanes of dual polar spaces, or all hyperplanes of certain dual polar spaces, arise from embeddings.     

On symplectic polar spaces over non-perfect fields of characteristic 2

Given a field 𝕂 of characteristic 2 and an integer n ≥ 2, let W(2n ? 1, 𝕂) be the symplectic polar space defined in PG(2n ? 1, 𝕂) by a non-degenerate alternating form of V(2n, 𝕂) and let Q(2n, 𝕂) be the quadric of PG(2n, 𝕂) associated to a non-singular quadratic form of Witt index n. In the literature it is often claimed that W(2n ? 1, 𝕂) ? Q(2n, 𝕂). This is true when 𝕂 is perfect, but false otherwise. In this article, we modify the previous claim in order to obtain a statement that is correct for any field of characteristic 2. Explicitly, we prove that W(2n ? 1, 𝕂) is indeed isomorphic to a non-singular quadric Q, but when 𝕂 is non-perfect the nucleus of Q has vector dimension greater than 1. So, in this case, Q(2n, 𝕂) is a proper subgeometry of W(2n ? 1, 𝕂). We show that, in spite of this fact, W(2n ? 1, 𝕂) can be embedded in Q(2n, 𝕂) as a subgeometry and that this embedding induces a full embedding of the dual DW(2n ? 1, 𝕂) of W(2n ? 1, 𝕂) into the dual DQ(2n, 𝕂) of Q(2n, 𝕂).     

Valuations and hyperplanes of dual polar spaces

Valuations were introduced in De Bruyn and Vandecasteele (Valuations of near polygons,preprint, 2004) as a very important tool for classifying near polygons. In the present paper we study valuations of dual polar spaces. We will introduce the class of the SDPS-valuations and characterize these valuations. We will show that a valuation of a finite thick dual polar space is the extension of an SDPS-valuation if and only if no induced hex valuation is ovoidal or semi-classical. Each SDPS-valuation will also give rise to a geometric hyperplane of the dual polar space.     

On geometric SDPS-sets of elliptic dual polar spaces

Let n∈N?{0,1} and let K and K^′ be fields such that K^′ is a quadratic Galois extension of K. Let Q⁻(2n+1,K) be a nonsingular quadric of Witt index n in PG(2n+1,K) whose associated quadratic form defines a nonsingular quadric Q⁺(2n+1,K^′) of Witt index n+1 in PG(2n+1,K^′). For even n, we define a class of SDPS-sets of the dual polar space DQ⁻(2n+1,K) associated to Q⁻(2n+1,K), and call its members geometric SDPS-sets. We show that geometric SDPS-sets of DQ⁻(2n+1,K) are unique up to isomorphism and that they all arise from the spin embedding of DQ⁻(2n+1,K). We will use geometric SDPS-sets to describe the structure of the natural embedding of DQ⁻(2n+1,K) into one of the half-spin geometries for Q⁺(2n+1,K^′).     

On the simple connectedness of hyperplane complements in dual polar spaces, II

Suppose Δ is a dual polar space of rank n and H is a hyperplane of Δ. Cardinali, De Bruyn and Pasini have already shown that if n≥4 and the line size is greater than or equal to 4 then the hyperplane complement Δ−H is simply connected. This paper is a follow-up, where we investigate the remaining cases. We prove that the hyperplane complements are simply connected in all cases except for three specific types of hyperplane occurring in the smallest case, when the rank and the line size are both 3.     

Metric characterization of apartments in dual polar spaces

Let Π be a polar space of rank n and let Gk(Π), k∈{0,…,n−1} be the polar Grassmannian formed by k-dimensional singular subspaces of Π. The corresponding Grassmann graph will be denoted by Γk(Π). We consider the polar Grassmannian Gn−1(Π) formed by maximal singular subspaces of Π and show that the image of every isometric embedding of the n-dimensional hypercube graph Hn in Γn−1(Π) is an apartment of Gn−1(Π). This follows from a more general result concerning isometric embeddings of Hm, m?n in Γn−1(Π). As an application, we classify all isometric embeddings of Γn−1(Π) in Γn^′−1(Π^′), where Π^′ is a polar space of rank n^′?n.     

Minimal full polarized embeddings of dual polar spaces

Let Δ be a thick dual polar space of rank n ≥ 2 admitting a full polarized embedding e in a finite-dimensional projective space Σ, i.e., for every point x of Δ, e maps the set of points of Δ at non-maximal distance from x into a hyperplane e∗(x) of Σ. Using a result of Kasikova and Shult [11], we are able the show that there exists up to isomorphisms a unique full polarized embedding of Δ of minimal dimension. We also show that e∗ realizes a full polarized embedding of Δ into a subspace of the dual of Σ, and that e∗ is isomorphic to the minimal full polarized embedding of Δ. In the final section, we will determine the minimal full polarized embeddings of the finite dual polar spaces DQ(2n,q), DQ ⁻(2n+1,q), DH(2n−1,q ²) and DW(2n−1,q) (q odd), but the latter only for n≤ 5. We shall prove that the minimal full polarized embeddings of DQ(2n,q), DQ ⁻(2n+1,q) and DH(2n−1,q ²) are the `natural' ones, whereas this is not always the case for DW(2n−1, q).B. De Bruyn: Postdoctoral Fellow of the Research Foundation - Flanders.     

Two classes of hyperplanes of dual polar spaces without subquadrangular quads

Let Π be one of the following polar spaces: (i) a nondegenerate polar space of rank n−1?2 which is embedded as a hyperplane in Q(2n,K); (ii) a nondegenerate polar space of rank n?2 which contains Q(2n,K) as a hyperplane. Let Δ and DQ(2n,K) denote the dual polar spaces associated with Π and Q(2n,K), respectively. We show that every locally singular hyperplane of DQ(2n,K) gives rise to a hyperplane of Δ without subquadrangular quads. Suppose Π is associated with a nonsingular quadric Q⁻(2n+?,K) of PG(2n+?,K), ?∈{−1,1}, described by a quadratic form of Witt-index , which becomes a quadratic form of Witt-index when regarded over a quadratic Galois extension of K. Then we show that the constructed hyperplanes of Δ arise from embedding.     

On certain forms and quadrics related to symplectic dual polar spaces in characteristic 2

Let V be a 2n-dimensional vector space over a field ${\mathbb {F}}$ and ξ a non-degenerate alternating form defined on V. Let Δ be the building of type C _n formed by the totally ξ-isotropic subspaces of V and, for 1 ≤ k ≤ n, let ${\mathcal {G}_k}$ and Δ _k be the k-grassmannians of PG(V) and Δ, embedded in ${W_k=\wedge^kV}$ and in a subspace ${V_k\subseteq W_k}$ respectively, where ${{\rm dim}(V_k)={2n \choose k} - {2n \choose {k-2}}}$ . This paper is a continuation of Cardinali and Pasini (Des. Codes. Cryptogr., to appear). In Cardinali and Pasini (to appear), focusing on the case of k = n, we considered two forms α and β related to the notion of ‘being at non maximal distance’ in ${\mathcal {G}_n}$ and Δ _n and, under the hypothesis that ${{\rm char}(\mathbb {F}) \neq 2}$ , we studied the subspaces of W _n where α and β coincide or are opposite. In this paper we assume that ${{\rm char}(\mathbb {F}) = 2}$ . We determine which of the quadrics associated to α or β are preserved by the group ${G= {\rm Sp}(2n, \mathbb {F})}$ in its action on W _n and we study the subspace ${\mathcal {D}}$ of W _n formed by vectors v such that α(v, x) = β(v, x) for every ${x \in W_n}$ . Finally, we show how properties of ${\mathcal {D}}$ can be exploited to investigate the poset of G-invariant subspaces of V _k for k = n ? 2i and ${1\leq i \leq \lfloor n/2\rfloor}$ .     

Locally singular hyperplanes in thick dual polar spaces of rank 4

We study (i-)locally singular hyperplanes in a thick dual polar space Δ of rank n. If Δ is not of type DQ(2n,K), then we will show that every locally singular hyperplane of Δ is singular. We will describe a new type of hyperplane in DQ(8,K) and show that every locally singular hyperplane of DQ(8,K) is either singular, the extension of a hexagonal hyperplane in a hex or of the new type.