Quasi-subgeometry partitions of projective spaces

Abstract

The notion of a subgeometry partition of a finite projective space $% PG(2m-1,q^{2})$ by $PG(m-1,q^{2})$'s and $PG(2m-1,q)$'s or a partition of $% PG(2m,q^{2})$ by $PG(2m,q)$'s is generalized to quasi-subgeometry partitions of $PG(2m-1,q^{d})$ by $PG(dm/e-1,q^{e})$'s for a set of divisors $e$ of $d$ and, partitions of $PG(2m,q^{2d})$ by $PG(d(2m+1)/f-1,q^{f})$'s for a set of divisors $f$ of $d$. In all cases, there are associated vector space spreads that are unions of `fans'.

More generally, in the arbitrary dimensional case, a complete theory of quasi-subgeometry partitions of $PG(V-1,D)$ corresponding to generalized spreads admitting $D^{\ast }$ as a fixed-point-free collineation group is obtained. When $D$ is a quadratic extension of a base field, `subgeometry' partitions are obtain.