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

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)

Hyperplanes of symplectic dual polar spaces : a survey

A

The hyperplanes of the U (4)(3) near hexagon

Combining theoretical arguments with calculations in the computer algebra package GAP, we are able to show that there are 27 isomorphism classes of hyperplanes in the near hexagon for the group U (4)(3). We give an explicit construction of a representative of each class and we list several combinatorial properties of such a representative

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

Non-classical hyperplanes of DW(5, q)

The hyperplanes of the symplectic dual polar space DW(5, q) arising from embedding, the so-called classical hyperplanes of DW(5, q), have been determined earlier in the literature. In the present paper, we classify non-classical hyperplanes of DW(5, q). If q is even, then we prove that every such hyperplane is the extension of a non-classical ovoid of a quad of DW(5, q). If q is odd, then we prove that every non-classical ovoid of DW(5, q) is either a semi-singular hyperplane or the extension of a non-classical ovoid of a quad of DW(5, q). If DW(5, q), q odd, has a semi-singular hyperplane, then q is not a prime number

On the simple connectedness of hyperplane complements in dual polar spaces

Let $\Delta$ be a dual polar space of rank \geq 4$,$ be a hyperplane of $\Delta$ and $\Gamma: = \Delta\setminus H$ be the complement of $in$\Delta$. We shall prove that, if all lines of$\Delta$have more than$ points, then $\Gamma$ 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

The hyperplanes of DQ(-)(7,K) arising from embedding

AbstractWe determine all hyperplanes of the dual polar space DQ−(7,K) which arise from embedding. This extends one of the results of [B. De Bruyn. The hyperplanes of DQ(2n,K) and DQ−(2n+1,q) which arise from their spin-embeddings, J. Combin. Theory Ser. A 114 (2007), 681–691] to the infinite case