On some subvarieties of the Grassmann variety

Let  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  with non-empty intersection with Π. It is known that the Pl\"{u}cker embedding of the elements of  is a variety of PG(rt−1,q), say rt. In this paper, we describe the image under the Pl\"{u}cker 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 rt.