m-Systems and Partial m-Systems of Polar Spaces

Abstract

Let P be a finite classical polar space of rank r, with r ≥ 2. A partial m-system M of P, with 0 ≤ m ≤ r - 1, is any set (π ₁, π ₂,..., π _k } of k (≠ 0) totally singular m-spaces of P such that no maximal totally singular space containing π _i has a point in common with (π ₁ ∪ π ₂ ∪... ∪π _k ) - π _i , i = 1, 2,..., k. In a previous paper an upper bound δ for ∣M∣ was obtained (Theorem 1). If ∣M∣ = δ, then M is called an m-system of P. For m = 0 the m-systems are the ovoids of P; for m = r - 1 the m-systems are the spreads of P. In this paper we improve in many cases the upper bound for the number of elements of a partial m-system, thus proving the nonexistence of several classes of m-systems.