Tight sets and \(m\)-ovoids of finite polar spaces. (English) Zbl 1124.51003
Summary: An intriguing set of points of a generalised quadrangle was introduced in [J. Bamberg, M. Law, and 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 [loc. cit.] 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, Geom. Dedicata 10 135–143 (1981; Zbl 0458.51010)] 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, Luycks, and Thas [S. De Winter and J. A. Thas Des. Codes Cryptography 32, No. 1–3, 153–166 (2004; Zbl 1056.51006), D. Luyckx, \(m\)-systems of finite classical polar spaces, PhD thesis, Univ. 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.
MSC:
| 51E20 | Combinatorial structures in finite projective spaces |
| 51A50 | Polar geometry, symplectic spaces, orthogonal spaces |
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