B. De Bruyn

On Linear Representations of Near Hexagons

We discuss linear representations of near polygons in affine spaces. All linear representations of near hexagons in an affine space of orderq? 3 and dimension up to seven are classified. If the dimension of the affine space is at least eight, then the near hexagon necessarily contains a quad of typeT*2(O) and every such quad has a rosette of ovoids. We conjecture that there are no such examples.

On the simple connectedness of hyperplane complements in dual polar spaces, II

Suppose @D is a dual polar space of rank n and H is a hyperplane of @D. Cardinali, De Bruyn and Pasini have already shown that if n>=4 and the line size is greater than or equal to 4 then the hyperplane complement @D-H is simply connected. This paper is a follow-up, where we investigate the remaining cases. We prove that the hyperplane complements are simply connected in all cases except for three specific types of hyperplane occurring in the smallest case, when the rank and the line size are both 3.

Isometric full embeddings of DW (2n-1,q) into DH(2n-1,q2)

We show that there is up to isomorphism a unique isometric full embedding of the dual polar space DW(2n-1,q) into the dual polar space DH(2n-1,q^2). We use the theory of valuations of near polygons to study the structure of this isometric embedding. We show that for every point x of DH(2n-1,q^2) at distance @d from DW(2n-1,q) the set of points of DW(2n-1,q) at distance @d from x is a so-called SDPS-set which carries the structure of a dual polar space DW(2@d-1,q^2). We show that if n is even, then the set of points at distance at most n2-1 from DW(2n-1,q) is a geometric hyperplane of DH(2n-1,q^2) and we study some properties of these new hyperplanes.

Locally singular hyperplanes in thick dual polar spaces of rank 4

We study (i-)locally singular hyperplanes in a thick dual polar space Δ of rank n. If Δ is not of type DQ(2n, K), then we will show that every locally singular hyperplane of Δ 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 a hex or of the new type.

On symplectic polar spaces over non-perfect fields of characteristic 2

Given a field 𝕂 of characteristic 2 and an integer n ≥ 2, let W(2n − 1, 𝕂) be the symplectic polar space defined in PG(2n − 1, 𝕂) by a non-degenerate alternating form of V(2n, 𝕂) and let Q(2n, 𝕂) be the quadric of PG(2n, 𝕂) associated to a non-singular quadratic form of Witt index n. In the literature it is often claimed that W(2n − 1, 𝕂) ≅ Q(2n, 𝕂). This is true when 𝕂 is perfect, but false otherwise. In this article, we modify the previous claim in order to obtain a statement that is correct for any field of characteristic 2. Explicitly, we prove that W(2n − 1, 𝕂) is indeed isomorphic to a non-singular quadric Q, but when 𝕂 is non-perfect the nucleus of Q has vector dimension greater than 1. So, in this case, Q(2n, 𝕂) is a proper subgeometry of W(2n − 1, 𝕂). We show that, in spite of this fact, W(2n − 1, 𝕂) can be embedded in Q(2n, 𝕂) as a subgeometry and that this embedding induces a full embedding of the dual DW(2n − 1, 𝕂) of W(2n − 1, 𝕂) into the dual DQ(2n, 𝕂) of Q(2n, 𝕂).

On the trivectors of a 6-dimensional symplectic vector space

Let V be a 6-dimensional vector space over a field F, let f be a nondegenerate alternating bilinear form on V and let Sp(V, f) ∼= Sp6(F) denote the symplectic group associated with (V, f). The group GL(V ) has a natural action on the third exterior power ∧3 V of V and this action defines five families of nonzero trivectors of V . Four of these families are orbits for any choice of the field F. The orbits of the fifth family are in one-to-one correspondence with the quadratic extensions of F that are contained in a fixed algebraic closure F of F. In this paper, we divide the orbits corresponding to the separable quadratic extensions into suborbits for the action of Sp(V, f) ⊆ GL(V ) on ∧3 V .

On Small and Large Hyperplanes of DW (5, q )

We determine lower and upper bounds for the size of a hyperplane of the dual polar space DW(5, q). In some cases, we also determine all hyperplanes attaining these bounds.

The simple connectedness of hyperplane complements in thick dual polar spaces of rank at least 4

Abstract Let Δ be a dual polar space of rank n ⩾ 4 , H be a hyperplane of Δ and Γ : = Δ \ H be the complement of H in Δ. We shall prove that, if all lines of Δ 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 spaces, arise from embeddings.