  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 ${\mathcal V}_{rt}$.
