m-Systems and Partial m-Systems of Polar Spaces

E. E. Shult and J. A. Thas
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E. E. Shult: Kansas State University, Department of Mathematics
J. A. Thas: University of Ghent, Department of Pure Mathematics and Computer Algebra

A chapter in Designs and Finite Geometries, 1996, pp 229-238 from Springer

Abstract: 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.

Keywords: m-system; partial m-system; polar space (search for similar items in EconPapers)
Date: 1996
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DOI: 10.1007/978-1-4613-1395-3_17

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