Segre Varieties and Desarguesian Spreads

TL;DR

This paper explores the intersection properties of Desarguesian spreads in projective spaces, linking geometric configurations with algebraic structures over extension fields and introducing generalized Segre varieties.

Contribution

It introduces a new matrix model for Desarguesian spreads and characterizes their intersections via subgeometries over extension fields, advancing the understanding of their geometric and algebraic structure.

Findings

Intersection of two Desarguesian spreads is determined by a subgeometry over an extension field.

Maximal subspaces of generalized Segre varieties are characterized geometrically.

If two spreads share a pseudo-arc of size k+1, their intersection is a specific system of subspaces.

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 $\mathcal S^r_{kr-1,h-1}(q)$ 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 $\mathcal R^r_{h,q}$ of $(h-1)$-dimensional subspaces of $\mathcal S^r_{kr-1,h-1}(q)$, for some proper divisor $r$ of $h$.