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

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

Hyperplanes of symplectic dual polar spaces : a survey

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On hyperovals of polar spaces

We derive lower and upper bounds for the size of a hyperoval of a finite polar space of rank 3. We give a computer-free proof for the uniqueness, up to isomorphism, of the hyperoval of size 126 of H(5, 4) and prove that the near hexagon E-3 has up to isomorphism a unique full embedding into the dual polar space DH(5, 4)

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