On some subvarieties of the Grassmann variety

Let S be a Desarguesian (t-1)-spread of PG(rt-1,q), $\Pi$. It is known that the Plucker embedding of the elements of S is a variety of PG(r^t-1,q), say V_{rt}. In this paper, we describe the image under the Plucker embedding of the elements of a linear set of PG(rt-1,q) of rank m 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 V_{rt}