Weighted intriguing sets of finite generalised quadrangles

Abstract

We construct and analyse interesting integer valued functions on the points of a generalised quadrangle which lie in the orthogonal complement of a principal eigenspace of the collinearity relation. These functions generalise the intriguing sets introduced by Bamberg et al. (Combinatorica 29(1):1-17, 2009), and they provide the extra machinery to give new proofs of old results and to establish new insight into the existence of certain configurations of generalised quadrangles. In particular, we give a geometric characterisation of Payne's tight sets, we give a new proof of Thas' result that an m-ovoid of a generalised quadrangle of order (s, s 2 ) is a hemisystem, and we give a bound on the values of m for which it is possible for an m-ovoid of the four dimensional Hermitian variety to exist.

Perfect 2-Colorings of the Grassmann Graph of Planes

“…Tight sets and intriguing sets in the geometric setting cover a variety of topics of interest to finite geometers, including movoids of generalized quadrangles and Cameron-Liebler classes among others. Intriguing sets, with a focus on tight sets, in (the point graph of) various finite geometries have received a lot of attention in recent years; see for example [1,6,7,11].…”

New Non-existence Proofs for Ovoids of Hermitian Polar Spaces and Hyperbolic Quadrics

“…The assumption says that there are 3 noncollinear points of the ovoid in the 14th orbit. All orbits except 2,4,8,10,12,13,14,15,16,18,19,25,26,32 contain no point of the ovoid, since their points are in the perp of one point of the 14th orbit. The orbits 16,18,25,32 are empty, since they are subsets of W 5 and all 3 points of the ovoid in W 5 are in the 14th orbit.…”

On regular systems of finite classical polar spaces

“…is an ovoid, and {O Z | Z ∈ t} is a fan. Finally, up to date, no infinite family of m-regular systems of H(4, q 2 ) is known, although several examples have been found when q ∈ {2, 3, 4, 5} [2].…”

“…If k = 1 and the rank d of the polar space is at least three, then some constructions are known on the parabolic quadric Q(2d, q) [18,41] and on the Hermitian variey H(2d − 1, q) [10,20]. Finally since polar spaces of rank two are generalized quadrangles, several examples of regular systems arise from m-ovoids by using duality, see [2,4,8,7,15,16,18,19,33,34,40,44,43]. However, in the case of generalized quadrangles, many questions are still unsolved.…”