On some subvarieties of the Grassmann variety

TL;DR

This paper characterizes the algebraic variety formed by certain linear sets in projective space, showing they are projections of Veronese varieties and linear sections of Grassmannian embeddings.

Contribution

It describes the Plücker image of linear sets intersecting a subspace as projections of Veronese varieties and linear sections of Grassmannian varieties.

Findings

The image under Plücker embedding is an algebraic variety.

This variety is a projection of a Veronese variety.

It is a linear section of the Grassmannian variety.

Abstract

Let $\mathcal S$ be a Desarguesian $(t-1)$--spread of $PG(rt-1,q)$, $\Pi$ a $m$-dimensional subspace of $PG(rt-1,q)$ and $\Lambda$ the linear set consisting of the elements of $\mathcal S$ with non-empty intersection with $\Pi$. It is known that the Pl\"{u}cker embedding of the elements of $\mathcal S$ is a variety of $PG(r^t-1,q)$, say ${\mathcal V}_{rt}$. In this paper, we describe the image under the Pl\"{u}cker embedding of the elements of $\Lambda$ and we show that it is an $m$-dimensional algebraic variety, projection of a Veronese variety of dimension $m$ and degree $t$, and it is a suitable linear section of ${\mathcal V}_{rt}$.