170 research outputs found
Locally subquadrangular hyperplanes in symplectic and Hermitian dual polar spaces
AbstractIn Pasini and Shpectorov (2001)Â [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
The hyperplanes of which arise from embedding.
We show that there are 6 isomorphism classes of hyperplanes of the dual polar space which arise from the Grassmann-embedding. If \geq 2\Delta, 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
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
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)
On the simple connectedness of hyperplane complements in dual polar spaces
Let be a dual polar space of rank \geq 4 be a hyperplane of
and be the complement of \Delta\Delta points, then 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
Non-classical hyperplanes of finite thick dual polar spaces
We obtain a classification of the non-classical hyperplanes of all finite thick dual polar spaces of rank at least 3 under the assumption that there are no ovoidal and semi-singular hex intersections. In view of the absence of known examples of ovoids and semi-singular hyperplanes in finite thick dual polar spaces of rank 3, the condition on the nonexistence of these hex intersections can be regarded as not very restrictive. As a corollary, we also obtain a classification of the non-classical hyperplanes of DW(2n - 1, q), q even. In particular, we obtain a complete classification of all non-classical hyperplanes of DW(2n - 1, q) if q is an element of {8, 32}
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