Segre Varieties and Desarguesian Spreads

Abstract

Let $\mathrm{PG}(n-1,q)$ denote the $(n-1)$-dimensional projective space over ${\mathbb{F}}_q$. We investigate the intersection of two Desarguesian $(h-1)$-spreads of $\mathrm{PG}(kh-1,q)$ and show that it is determined by a subgeometry over a suitable extension field. Our approach combines a characterization of subsets of points of $\mathrm{PG}(k-1,q^h)$ closed under $q$-order subgeometries with a matrix model for Desarguesian spreads based on Moore matrices. This leads naturally to the notion of generalized Segre varieties ${\mathcal{S}}_{kr-1,h-1}^r(q)$ and a geometric description of their maximal subspaces. As a main application, we prove that if two distinct Desarguesian $(h-1)$-spreads of $\mathrm{PG}(kh-1,q)$ contain a common pseudo-arc of size $k+1$, then their intersection is precisely the system ${\mathcal{R}}_{h,q}^r$ of $(h-1)$-dimensional subspaces of ${\mathcal{S}}_{kr-1,h-1}^r(q)$, for some proper divisor $r$ of $h$.