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

AbstractWe 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

Hyperplanes of symplectic dual polar spaces : a survey

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The hyperplanes of DW(5,2h)DW(5,2^h) which arise from embedding.

We show that there are 6 isomorphism classes of hyperplanes of the dual polar space $\Delta = DW(5,2^h)$ which arise from the Grassmann-embedding. If \geq 2$, then these are all the hyperplanes of$\Delta$arising from an embedding. If = 1$, then there are 6 extra classes of hyperplanes as has been shown by Pralle with the aid of a computer. We will give a computer free proof for this fact. The hyperplanes of (5,q)$,$ odd, arising from an embedding will be classified in the forthcoming paper

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 Delta of rank n >= 2: (i) The Grassmann-embedding of the symplectic dual polar space Delta congruent to DW(2n 1,q); (ii) the Grassmann-embedding of the Hermitian dual polar space Delta congruent to DH(2n-1, q(2)); (iii) the spin-embedding of the orthogonal dual polar space Delta congruent to DQ(2n, q); (iv) the spin-embedding of the orthogonal dual polar space Delta congruent to DQ(-)(2n+ 1, q). Let H-e denote the set of all hyperplanes of Delta arising from the embedding e. We give a method for constructing the hyperplanes of H-e without implementing the embedding e and discuss (possible) applications of the given construction

On a class of hyperplanes of the symplectic and Hermitian dual polar spaces.

Let $\Delta$ be a symplectic dual polar space (2n-1,K)$or a Hermitian dual polar space (2n-1,K,\theta)$, \geq 2$. We define a class of hyperplanes of$\Delta$arising from its Grassmann-embedding and discuss several properties of these hyperplanes. The construction of these hyperplanes allows us to prove that there exists an ovoid of the Hermitian dual polar space (2n-1,K,\theta)$ arising from its Grassmann-embedding if and only if there exists an empty $\theta variety in$\PG(n-1,K)$. Using this result we are able to give the first examples of ovoids in thick dual polar spaces of rank at least 3 which arise from some projective embedding. These are also the first examples of ovoids in thick dual polar spaces of rank at least 3 for which the construction does not make use of transfinite recursion

On extensions of hyperplanes of dual polar spaces

AbstractLet Δ be a thick dual polar space and F a convex subspace of diameter at least 2 of Δ. Every hyperplane G of the subgeometry F˜ of Δ induced on F will give rise to a hyperplane H of Δ, the so-called extension of G. We show that F and G are in some sense uniquely determined by H. We also consider the following problem: if e is a full projective embedding of Δ and if eF is the full embedding of F˜ induced by e, does the fact that G arises from the embedding eF imply that H arises from the embedding e? We will study this problem in the cases that e is an absolutely universal embedding, a minimal full polarized embedding or a Grassmann embedding of a symplectic dual polar space. Our study will allow us to prove that if e is absolutely universal, then also eF is absolutely universal

Characterizations of symplectic polar spaces

A polar space S is said to be symplectic if it admits an embedding e in a projective geometry PG(V) such that the e-image e(S) of S is defined by an alternating form of V. In this paper we characterize symplectic polar spaces in terms of their incidence properties, with no mention of peculiar properties of their embeddings. This is relevant especially when S admits different (non isomorphic) embeddings, as it is the case (precisely) when S is defined over a field of characteristic 2.Comment: 20 pages/extensively revise