Tight sets and m-ovoids of finite polar spaces

Abstract

An intriguing set of points of a generalised quadrangle was introduced in [J. Bamberg, M. Law, T. Penttila, Tight sets and m-ovoids of generalised quadrangles, Combinatorica, in press] as a unification of the pre-existing notions of tight set and m-ovoid. It was shown in [J. Bamberg, M. Law, T. Penttila, Tight sets and m-ovoids of generalised quadrangles, Combinatorica, in press] that every intriguing set of points in a finite generalised quadrangle is a tight set or an m-ovoid (for some m). Moreover, it was shown that an m-ovoid and an i-tight set of a common generalised quadrangle intersect in mi points. These results yielded new proofs of old results, and in this paper, we study the natural analogue of intriguing sets in finite polar spaces of higher rank. In particular, we use the techniques developed in this paper to give an alternative proof of a result of Thas [J.A. Thas, Ovoids and spreads of finite classical polar spaces, Geom. Dedicata 10 (1-4) (1981) 135-143] that there are no ovoids of H(2r, q 2 ), Q − (2r + 1, q), and W(2r − 1, q) for r > 2. We also strengthen a result of Drudge on the non-existence of tight sets in W(2r − 1, q), H(2r + 1, q 2 ), and Q + (2r + 1, q), and we give a new proof of a result of De Winter, Luyckx, and Thas [S. De Winter, J.A. Thas, SPG-reguli satisfying the polar property and a new semipartial geometry, Des. Codes Cryptogr. 32 (1-3) (2004) 153-166; D. Luyckx, m-Systems of finite classical polar spaces, PhD thesis, The University of Ghent, 2002] that an m-system of W(4m + 3, q) or Q − (4m + 3, q) is a pseudo-ovoid of the ambient projective space.

Intriguing Sets of Vertices of Regular Graphs

“…These point-line geometries include the generalized quadrangles [5,20,21], the polar spaces [4,11], certain half-spin geometries [10] and the partial quadrangles [2]. These papers mainly deal with the construction and classification (sometimes with the aid of a computer) of intriguing sets, as well as the derivation of some of their properties (in the style of Propositions 3.4, 3.7, 3.8 and Corollaries 3.6, 3.12 below).…”

Intriguing Sets of Points of Q(2n, 2) \Q +(2n − 1, 2)

“…The following propositions are extracted from [8] (Propositions 3.4, 3.7 and Corollaries 3.9, 3.12). Proofs of Propositions 2.1, 2.2 and 2.3 are also more or less contained in the earlier literature on intriguing sets ( [2], [3], [7], [11]). Proposition 2.1 Let X 1 and X 2 be two intriguing sets of vertices of Γ of the same index i ∈ {1, .…”

“…Intriguing sets of vertices have been studied for several classes of strongly regular graphs which arise as collinearity graphs of point-line geometries, see [3], [11] and [12] for generalized quadrangles, [2] for polar spaces, [7] for half-spin geometries and [1] for partial quadrangles. In the present paper, we study the intriguing sets of another class of strongly regular graphs.…”

Regular partitions of half-spin geometries

“…In the sequel, we list nontrivial examples of regular partitions of HS(2n − 1, q), hereby assuming that n ≥ 4 so that HS(2n − 1, q) is not a linear space. We also note that if n ∈ {4, 5}, then Γ is a strongly regular graph (the collinearity graph of Q + (7, q) if n = 4) and regular partitions with only two parts were already studied for these geometries, see [1] and [4].…”