Non-classical hyperplanes of DW(5, q)

The hyperplanes of the symplectic dual polar space DW(5, q) arising from embedding, the so-called classical hyperplanes of DW(5, q), have been determined earlier in the literature. In the present paper, we classify non-classical hyperplanes of DW(5, q). If q is even, then we prove that every such hyperplane is the extension of a non-classical ovoid of a quad of DW(5, q). If q is odd, then we prove that every non-classical ovoid of DW(5, q) is either a semi-singular hyperplane or the extension of a non-classical ovoid of a quad of DW(5, q). If DW(5, q), q odd, has a semi-singular hyperplane, then q is not a prime number

Points and hyperplanes of the universal embedding space of the dual polar space DW(5,q), q odd

It was proved earlier that there are 6 isomorphism classes of hyperplanes in the dual polar space (5,q)$,$ even, which arise from its Grassmann-embedding. In the present paper, we extend these results to the case that $is odd. Specifically, we determine the orbits of the full automorphism group of (5,q)$, $odd, on the projective points (or equivalently, the hyperplanes) of the projective space (13,q)$ which affords the universal embedding of (5,q)$

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

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}

Hyperplanes of symplectic dual polar spaces : a survey

A

Incidence geometry from an algebraic graph theory point of view

The goal of this thesis is to apply techniques from algebraic graph theory to finite incidence geometry. The incidence geometries under consideration include projective spaces, polar spaces and near polygons. These geometries give rise to one or more graphs. By use of eigenvalue techniques, we obtain results on these graphs and on their substructures that are regular or extremal in some sense. The first chapter introduces the basic notions of geometries, such as projective and polar spaces. In the second chapter, we introduce the necessary concepts from algebraic graph theory, such as association schemes and distance-regular graphs, and the main techniques, including the fundamental contributions by Delsarte. Chapter 3 deals with the Grassmann association schemes, or more geometrically: with the projective geometries. Several examples of interesting subsets are given, and we can easily derive completely combinatorial properties of them. Chapter 4 discusses the association schemes from classical finite polar spaces. One of the main applications is obtaining bounds for the size of substructures known as partial m- systems. In one specific case, where the partial m-systems are partial spreads in the polar space H(2d − 1, q^2) with d odd, the bound is new and even tight. A variant of the famous Erdős-Ko-Rado problem is considered in Chapter 5, where we study sets of pairwise non-trivially intersecting maximal totally isotropic subspaces in polar spaces. A combination of geometric and algebraic techniques is used to obtain a classification of such sets of maximum size, except for one specific polar space, namely H(2d − 1, q^2) for odd rank d ≥ 5. Near polygons, including generalized polygons and dual polar spaces, are studied in the last chapter. Several results on substructures in these geometries are given. An inequality of Higman on the parameters of generalized quadrangles is generalized. Finally, it is proved that in a specific dual polar space, a highly regular substructure would yield a distance- regular graph, generalizing a result on hemisystems. The appendix consists of an alternative proof for one of the main results in the thesis, a list of open problems and a summary in Dutch

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)

Implications of vanishing Krein parameters on Delsarte designs, with applications in finite geometry

In this paper we show that if $\theta$ is a $T$-design of an association scheme $(\Omega, \mathcal{R})$, and the Krein parameters $q_{i,j}^h$ vanish for some $h \not \in T$ and all $i, j \not \in T$ ($i, j, h \neq 0$), then $\theta$ consists of precisely half of the vertices of $(\Omega, \mathcal{R})$ or it is a $T'$-design, where $|T'|>|T|$. We then apply this result to various problems in finite geometry. In particular, we show for the first time that nontrivial $m$-ovoids of generalised octagons of order $(s, s^2)$ do not exist. We give short proofs of similar results for (i) partial geometries with certain order conditions; (ii) thick generalised quadrangles of order $(s,s^2)$; (iii) the dual polar spaces $\mathsf{DQ}(2d, q)$, $\mathsf{DW}(2d-1,q)$ and $\mathsf{DH}(2d-1,q^2)$, for $d \ge 3$; (iv) the Penttila-Williford scheme. In the process of (iv), we also consider a natural generalisation of the Penttila-Williford scheme in $\mathsf{Q}^-(2n-1, q)$, $n\geqslant 3$.Comment: This paper builds on part of the doctoral work of the second author under the supervision of the first. The second author acknowledges the support of an Australian Government Research Training Program Scholarship and Australian Research Council Discovery Project DP20010195