Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: On a class of hyperplanes of the symplectic and Hermitian dual polar spaces. | On a class of hyperplanes of the symplectic and Hermitian dual polar spaces Bart De Bruyn Department of Pure Mathematics and Computer Algebra Ghent University Gent Belgium bdb@ Submitted Jan 9 2008 Accepted Dec 15 2008 Published Jan 7 2009 Mathematics Subject Classihcations 51A45 51A50 Abstract Let A be a symplectic dual polar space DW 2n 1 K or a Hermitian dual polar space DH 2n 1 K Ớ n 2. We dehne a class of hyperplanes of A arising from its Grassmann-embedding and discuss several properties of these hyperplanes. The construction of these hyperplanes allows us to prove that there exists an ovoid of the Hermitian dual polar space DH 2n 1 K Ỡ arising from its Grassmann-embedding if and only if there exists an empty Ớ-Hermitian variety in PG n 1 K . Using this result we are able to give the hrst examples of ovoids in thick dual polar spaces of rank at least 3 which arise from some projective embedding. These are also the hrst examples of ovoids in thick dual polar spaces of rank at least 3 for which the construction does not make use of transhnite recursion. 1 Introduction Basic definitions Let n be a non-degenerate thick polar space of rank n 2. With n there is associated a point-line geometry A whose points are the maximal singular subspaces of n whose lines are the next-to-maximal singular subspaces of n and whose incidence relation is reverse containment. The geometry A is called a dual polar space Cameron 3 . There exists a bijective correspondence between the non-empty convex subspaces of A and the possibly empty singular subspaces of n if a is a singular subspace of n then the set of all maximal singular subspaces containing a is a convex subspace of A. If x and y are two points of A then d x y denotes the distance between x and y in the collinearity graph Postdoctoral Fellow of the Research Foundation - Flanders THE ELECTRONIC JOURNAL OF COMBINATORICS 16 2009 R1 1 of A. The maximal distance between two points of a convex subspace A of A is .
