Problems by the wayside
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 in it). The connection with the diagonal axiom is explained. These ideas are then used to provide characterization results for a certain class of semi-partial geometries, for symplectic hyperbolic line spaces, and for half Grassmann spaces. This is an extraordinary paper written by an extraordinary mathematician.
scientific article; zbMATH DE number 2182140
scientific article