On some subvarieties of the Grassmann variety

Let $\mathcal S$ be a Desarguesian $(t-1)$--spread of $PG(rt-1,q)$, $Π$ a $m$-dimensional subspace of $PG(rt-1,q)$ and $Λ$ the linear set consisting of the elements of $\mathcal S$ with non-empty intersection with $Π$. It is known that the Plücker 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 Plücker embedding of the elements of $Λ$ 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 ${\mathcal V}_{rt}$.

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