A characterization of m-ovoids and i-tight sets of polar spaces

Abstract

Abstract Let P be a finite polar space of rank r ≥ 2 with q + 1 ≥ 3 points on each line. In [J. Bamberg, S. Kelly, M. Law, T. Penttila, Tight sets and m-ovoids of finite polar spaces. J. Combin. Theory Ser. A 114 (2007), 1293–1314. MR2353124] and [J. Bamberg, M. Law, T. Penttilla, Tight sets and m-ovoids of generalised quadrangles. Combinatorica, to appear.] it was shown that every m-ovoid of P intersects every i-tight set of P in precisely mi points. In the present paper, we characterize m-ovoids of P as those sets of points which have constant intersection size with each member of a “nice family” of i-tight sets of P and conversely, we characterize i-tight sets of P as those sets of points which have constant intersection size with each member of a “nice family” of m-ovoids. Some interesting corollaries of these characterization theorems are given.