On the algebraic variety Vr,t

The variety Vr,t is the image under the Grassmannian map of the (t−1)-subspaces of PG(rt−1,q) of the elements of a Desarguesian spread. We investigate some properties of this variety, with particular attention to the case r=2: in this case we prove that every t+1 points of the variety are in general position and we give a new interpretation of linear sets of PG(1,qt).