Hyperplanes Of Dual Polar Spaces Research Articles

The hyperplanes of finite symplectic dual polar spaces which arise from projective embeddings

We characterize the hyperplanes of the dual polar space DW(2n−1,q) which arise from projective embeddings as those hyperplanes H of DW(2n−1,q) which satisfy the following property: if Q is an ovoidal quad, then Q∩H is a classical ovoid of Q. A consequence of this is that all hyperplanes of the dual polar spaces DW(2n−1,4), DW(2n−1,16) and DW(2n−1,p) (p prime) arise from projective embeddings.

Direct Constructions of Hyperplanes of Dual Polar Spaces Arising from Embeddings

Let e be one of the following full projective embeddings of a finite dual polar space Δ of rank n ≥ 2: (i) The Grassmann-embedding of the symplectic dual polar space Δ ≅ DW(2n – 1, q); (ii) the Grassmann-embedding of the Hermitian dual polar space Δ ≅ DH(2n – 1, q 2); (iii) the spin-embedding of the orthogonal dual polar space Δ ≅ DQ(2n, q); (iv) the spin-embedding of the orthogonal dual polar space Δ ≅DQ −(2n + 1, q). Let \({\mathcal{H}_{e}}\) denote the set of all hyperplanes of Δ arising from the embedding e. We give a method for constructing the hyperplanes of \({\mathcal{H}_{e}}\) without implementing the embedding e and discuss (possible) applications of the given construction.

The uniqueness of the SDPS-set of the symplectic dual polar space [formula omitted], [formula omitted

SDPS-sets are very nice sets of points in dual polar spaces which themselves carry the structure of dual polar spaces. They were introduced in [B. De Bruyn, P. Vandecasteele, Valuations and hyperplanes of dual polar spaces, J. Combin. Theory Ser. A 112 (2005) 194–211] because they gave rise to new valuations and hyperplanes of dual polar spaces. In the present paper, we show that the symplectic dual polar space D W ( 4 n − 1 , q ) , n ≥ 2 , has up to isomorphisms a unique SDPS-set.

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.

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

The simple connectedness of hyperplane complements in thick dual polar spaces of rank at least 4

Abstract 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.

A characterization of the SDPS-hyperplanes of dual polar spaces

In [B. De Bruyn, P. Vandecasteele, Valuations and hyperplanes of dual polar spaces, J. Combin. Theory Ser. A 112 (2005) 194–211], we introduced the class of the SDPS-valuations of dual polar spaces. We showed that these valuations and all their extensions give rise to hyperplanes of dual polar spaces. We call these hyperplanes SDPS-hyperplanes. In the present paper, we show that a hyperplane H of a thick dual polar space is an SDPS-hyperplane if and only if every hex A not contained in H intersects H in either a singular hyperplane or the extension of an ovoid.

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.

Hyperplanes of dual polar spaces of rank 3 with no subquadrangular quad

Hyperplanes of dual polar spaces of rank 3 with no subquadrangular quad