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

Let F and F' be two fields such that F' is a quadratic Galois extension of F. If vertical bar F vertical bar >= 3, then we provide sufficient conditions for a hyperplane of the Hermitian dual polar space DH(5, F') to arise from the Grassmann embedding. We use this to give an alternative proof for the fact that all hyperplanes of DH(5, q(2)), q not equal 2, arise from the Grassmann embedding, and to show that every hyperplane of DH(5, F') that contains a quad Q is either classical or the extension of a non-classical ovoid of Q. We will also give a classification of the hyperplanes of DH(5, F') that contain a quad and arise from the Grassmann embedding

An alternative definition of the notion valuation in the theory of near polygons.

Valuations of dense near polygons were introduced in \cite{DB-Va:1}. A valuation of a dense near polygon $\mathcal{S}=(\mathcal{P},\mathcal{L},\mathrm{I})$ is a map $from the point-set$\mathcal{P}$of$\mathcal{S}$to the set$\N$of nonnegative integers satisfying very nice properties with respect to the set of convex subspaces of$\mathcal{S}$. In the present paper, we give an alternative definition of the notion valuation and prove that both definitions are equivalent. In the case of dual polar spaces and many other known dense near polygons, this alternative definition can be significantly simplified

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

Polygonal valuations

AbstractWe develop a valuation theory for generalized polygons similar to the existing theory for dense near polygons. This valuation theory has applications for the study and classification of generalized polygons that have full subpolygons as subgeometries

Hyperplanes of embeddable Grassmannians arise from embeddings: a short proof

In this note, we give an alternative and considerably shorter proof of a result of Shult stating that all hyperplanes of embeddable Grassmannians arise from projective embeddings

The pseudo-hyperplanes and homogeneous pseudo-embeddings of AG(n, 4) and PG(n, 4)

We determine all homogeneous pseudo-embeddings of the affine space AG(n, 4) and the projective space PG(n, 4). We give a classification of all pseudo-hyperplanes of AG(n, 4). We also prove that the two homogeneous pseudo-embeddings of the generalized quadrangle Q(4, 3) are induced by the two homogeneous pseudo-embeddings of AG(4, 4) into which Q(4, 3) is fully embeddable

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

The uniqueness of the SDPS-set of the symplectic dual polar space DW(4n−1,q)DW(4n-1,q), n≥2n \geq 2

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 \cite{DB-V:2} 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 (4n-1,q)$, \geq 2$, has up to isomorphisms a unique SDPS-set