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

The hyperplanes of DQ(2n, K) and DQ(-) (2n+1, q) which arise from their spin-embeddings

AbstractWe characterize the hyperplanes of the dual polar spaces DQ(2n,K) and DQ−(2n+1,q) which arise from their respective spin-embeddings. The hyperplanes of DQ(2n,K) which arise from its spin-embedding are precisely the locally singular hyperplanes of DQ(2n,K). The hyperplanes of DQ−(2n+1,q) which arise from its spin-embedding are precisely the hyperplanes H of DQ−(2n+1,q) which satisfy the following property: if Q is an ovoidal quad, then Q∩H is a classical ovoid of Q

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

A property of isometric mappings between dual polar spaces of type DQ(2n,K)

Let f be an isometric embedding of the dual polar space Delta = DQ(2n, K) into Delta' = DQ(2n, K'). Let P denote the point-set of Delta and let e' : Delta' -> Sigma' congruent to PG(2(n) - 1, K') denote the spin-embedding of Delta'. We show that for every locally singular hyperplane H of Delta, there exists a unique locally singular hyperplane H' of Delta' such that f(H) = f(P) boolean AND H'. We use this to show that there exists a subgeometry Sigma congruent to PG(2(n) - 1, K) of Sigma' such that: (i) e' circle f (x) is an element of Sigma for every point x of Delta; (ii) e := e' circle f defines a full embedding of Delta into Sigma, which is isomorphic to the spin-embedding of Delta

Hyperplanes of symplectic dual polar spaces : a survey

A

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

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

Locally subquadrangular hyperplanes in symplectic and Hermitian dual polar spaces

AbstractIn Pasini and Shpectorov (2001) [11] all locally subquadrangular hyperplanes of finite symplectic and Hermitian dual polar spaces were determined with the aid of counting arguments and divisibility properties of integers. In the present note we extend this classification to the infinite case. We prove that symplectic dual polar spaces and certain Hermitian dual polar spaces cannot have locally subquadrangular hyperplanes if their rank is at least three and their lines contain more than three points