Problems by the wayside
type: Item
instance of: scholarly article
title: Problems by the wayside
zbMATH Open document ID: 1080.51001
published in: Discrete Mathematics
publication date: 2005-06-23T00:00:00Z
review text: In this unusual but mighty paper, the author touches three different subjects in incidence geometry (mainly finite geometry) and discusses the interplay of several mathematical objects and theories. Let us review these three parts. In the first section, the central object is a locally Hermitian ovoid of the generalized quadrangle \(H(3,q^2)\), i.e., a set of \(1+q^3\) points of the nonsingular Hermitian surface \(H(3,q^2)\) in \(PG(3,q^2)\) lying on \(q^2\) nondegenerate lines through one fixed point of \(H(3,q^2)\). The author relates it to subsets of points of \(AG(2,q)\) whose secants avoid certain slopes; sets of \(q^2\) lines of \(PG(2,q^2)\) pairwise meeting off a Hermitian cone; locally Hermitian \(1\)-systems on \(Q^-(5,q)\) (so locally Hermitian spreads of that quadrangle); line spreads of \(PG(3,q)\) and translation planes of order \(q^2\) with kernel \(GF(q)\). Furthermore, the author discusses \((0,2)\)-Hermitian sets of \(H(3,q^2)\) and applications. In the second section, ovoids, spreads and \(m\)-systems are discussed. The main target geometries are the polar spaces here, but the author proposes to try to generalize to other interesting geometries, in particular geometries related to buildings of exceptional type. For this, Kleidman's view on the definition of an ovoid seems to be a good starting point. An interesting problem in that respect is how one could define \(m\)-systems, \(0\neq m\neq n-1\), in polar spaces of rank \(n\) using the Kleidman view (hence without including the size of the object in the definition). Furthermore, the author applies some thoughts to two specific problems: that of ovoidal hyperplanes in dual polar spaces (geometric hyperplanes which are ovoids), and disjoint ovoids of \(Q^+(7,q)\). In the third section, the diagonal axiom for finite geometries is the central item. The oriflamme phenomenon is discussed, and the author introduces blocks in certain geometries (subspaces generated by any two intersecting lines contained …
Mathematics Subject Classification ID: 51A50
Mathematics Subject Classification ID: 51E12
Mathematics Subject Classification ID: 51E20
Mathematics Subject Classification ID: 51E24
Mathematics Subject Classification ID: 51A45
zbMATH DE Number: 2182140
zbMATH Keywords: finite geometry
zbMATH Keywords: ovoids
zbMATH Keywords: polar spaces
zbMATH Keywords: diagonal axiom
author: E. E. Shult
reviewed by: Hendrik Van Maldeghem
MaRDI profile type: Publication
full work available at URL: j.disc.2004.04.046
OpenAlex ID: W2065063565
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DOI: 10.1016/J.DISC.2004.04.046
description: scientific article; zbMATH DE number 2182140