Let 2 be a finite thick dual polar space of rank 3. We say that a hyperplane H of 2 is locally singular (respectively, quadrangular or ovoidal) if H & Q is the perp of a point (resp. a subquadrangle or an ovoid) of Q for every quad Q of 2. If H is locally singular, quadrangular, or ovoidal, then we say that H is uniform. It is known that if H is locally… (More)
Let ∆ be a dual polar space of rank n ≥ 4, H be a hyperplane of ∆ and Γ := ∆ \ H be the complement of H in ∆. We shall prove that, if all lines of ∆ have more than 3 points, then Γ 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… (More)
We introduce the concept of parallelism in diagram geometry, we apply it to a new gluing concept that provides geometries of higher rank, we combine it with another recent extension procedure for geometries and collect many examples solving existence questions for geometries over specified diagrams.
In this paper we consider partial linear spaces containing a set of subspaces isomorphic to affine planes, such that the lines and these afline planes on a fixed point form a non-degenerate polar spaces of rank at least 2. We obtain a complete classification, provided that the rank is at least 3. The study of geometries on the absolute points of polarities… (More)
We study (i-)locally singular hyperplanes in a thick dual polar space of rank n. If is not of type DQ(2n, K), then we will show that every locally singular hyperplane of is singular. We will describe a new type of hyperplane in DQ(8, K) and show that every locally singular hyperplane of DQ(8, K) is either singular, the extension of a hexagonal hyperplane in… (More)
Cooperstein ,  proved that every finite symplectic dual polar space DW (2n− 1, q), q = 2, can be generated by 2n n − 2n n−2 points and that every finite Hermitian dual polar space DH(2n − 1, q 2), q = 2, can be generated by 2n n points. In the present paper, we show that these conclusions remain valid for symplectic and Hermitian dual polar spaces… (More)