Subregular spreads of \({\mathcal PG}(2n+1,q)\)
A spread of \(PG(2n+1,q)\) is a partition into \(q^{n+1}+1\) pairwise disjoint \(n\)-spaces. A spread is said to be regular if the regulus determined by any three spaces of the spread is composed entirely of elements of the spread. In \(PG(3,q)\), the lines which meet every line of a regulus \(R\) form another regulus \(R'\) (the opposite regulus). These reguli have exactly the same points, so that the lines of \(R'\) can replace those of \(R\) in a spread. A spread obtained from a regular spread by a sequence of such regulus reversals is called subregular. In this paper, the idea of a subregular spread is generalized to higher odd-dimensional finite projective spaces. Regulus reversal (which is obviously specific to 3-spaces) is replaced by ``hyperregulus reversal'', where a hyperregulus, as described by Bruck and Ostrom, is a set of \((q^{n+1}-1)/(q-1)\) \(n\)-spaces that partitions a certain type of surface. Such a surface may permit multiple partitions; replacing one by another in a spread is the desired hyperregulus reversal operation, and subregular spreads are defined by analogy with the three-dimensional case. Building on work of Bruck, it is shown that subregular spreads on \(PG(2n+1,q)\) can be constructed using pairwise disjoint covers in the associated circle geometry \(CG(n+1,q)\). This is used to construct subregular spreads in \(PG(5,q)\); and it is shown that the resulting spreads are non-André. The paper ends with a number of open problems about subregular spreads.
scientific article; zbMATH DE number 1229765