Tight sets and m-ovoids of finite polar spaces

An intriguing set of points of a generalised quadrangle was introduced in [2] as a unification of the pre-existing notions of tight set and m-ovoid. It was shown in [2] 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 [36] that there are no ovoids of H(2r, q2), 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 due to De Winter, Luyckx, and Thas [9,28] that an m-system of W(4m+3, q) or Q−(4m+3, q) is a pseudo-ovoid of the ambient projective space.