TY - JOUR

T1 - Tight sets and m-ovoids of finite polar spaces

AU - Bamberg, John

AU - Law, M.

AU - Kelly, S.A.

AU - Penttila, Tim

PY - 2007

Y1 - 2007

N2 - An intriguing set of points of a generalised quadrangle was introduced in [J. Bamberg, M. Law, T. Pentfila, 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. Perittila, 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 in-ovoid (for some in). 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, q2), 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. (C) 2007 Elsevier Inc. All rights reserved.

AB - An intriguing set of points of a generalised quadrangle was introduced in [J. Bamberg, M. Law, T. Pentfila, 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. Perittila, 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 in-ovoid (for some in). 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, q2), 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. (C) 2007 Elsevier Inc. All rights reserved.

U2 - 10.1016/j.jcta.2007.01.009

DO - 10.1016/j.jcta.2007.01.009

M3 - Article

VL - 114

SP - 1293

EP - 1314

JO - Journal of Combinatorial Theory Series A

JF - Journal of Combinatorial Theory Series A

SN - 0021-9800

IS - 7

ER -