Ovoids, spreads and m-systems of finite polar spaces: existence, nonexistence, constructions
by
J. A. Thas
Ghent University

Let P be a finite polar space of rank r >= 2. An ovoid O of P is a pointset of P, which has exactly one point in common with each generator of P, that is, with each maximal totally singular subspace of P. A spread S of P is a set of generators, which constitutes a partition of the pointset. It appears that |O| = |S| for any ovoid O and any spread S of any given polar space P; this common number will be denoted by \mu_P and will be called the ovoid number of the polar space P. Ovoids and spreads have many connections with and applications to projective planes, circle geometries, generalized polygons, strongly regular graphs, partial geometries, semipartial geometries, codes, designs. Existence and nonexistence results on ovoids and spreads, and also constructions, are due to Bader, Brouwer, Brown, Conway, Dye, Kantor, Kleidman, Lunardon, Lüneburg, Moorhouse, O'Keefe, Payne, Penttila, Royle, Shult, Thas, Tits, Wilson, ... . Whether or not a particular polar space contains an ovoid or spread can be a very hard problem, and many cases are still open.

In ``m-Systems of polar spaces'', J. Combin. Theory Ser. A, 68:184-204, 1994, E. E. Shult and J. A. Thas introduced partial m-systems and m-systems of polar spaces. A partial m-system of the finite polar space P of rank r, with r >= 2 and 0 <= m <= r-1, is any set { \pi₁, \pi₂, ... , \pi_k } of k ( =/= 0) totally singular m-spaces of P such that no generator containing \pi_i has a point in common with (\pi₁ \cup \pi₂ \cup ... \cup \pi_k)-\pi_i, with i = 1, 2, ... , k. A partial 0-system of size k is also called a partial ovoid, or a cap or a k-cap; a partial (r-1)-system is also called a partial spread. For any partial m-system M of P we have |M| <= \mu_P. If |M|=\mu_P, then the partial m-system M of P is called an m-system of P. For m=0, the m-system is an ovoid of P; for m=r-1, with r the rank of P, the m-system is a spread of P. The fact that |M|, with M any m-system of the polar space P, is independent of m gives us an explanation why an ovoid and a spread of a polar space P have equal size. Partial m-systems and m-systems have many connections with and applications to codes, graphs and several interesting classes of incidence geometries. Existence and nonexistence theorems, and constructions, for 0 < m < r-1, are due to Hamilton, Mathon, Quinn, Shult and Thas.

We say that a partial m-system M of the polar space P has the BLT property if and only if there is no line of P meeting three distinct members of M non-trivially. In the paper ``Constructions of polygons from buildings'', Proc. London Math. Soc., 71:397-440, 1995, by E. E. Shult and J. A. Thas, these particular partial m-systems play a crucial role. Whether or not a particular (partial) m-system possessing the BLT property exists is still open in many cases.

Here a survey of the known existence and nonexistence theorems will be given, also in the case of the nonclassical finite polar spaces of rank 2, that is, the nonclassical finite generalized quadrangles; also many interesting constructions will be given.

Date received: January 16, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cagg-03.