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 ( 2 r , q 2 ) , Q − ( 2 r + 1 , q ) , and W ( 2 r − 1 , q ) for r > 2 . We also strengthen a result of Drudge on the non-existence of tight sets in W ( 2 r − 1 , q ) , H ( 2 r + 1 , q 2 ) , and Q + ( 2 r + 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 ( 4 m + 3 , q ) or Q − ( 4 m + 3 , q ) is a pseudo-ovoid of the ambient projective space.