The variety of exterior powers of linear maps

Let V and W be vector spaces of dimension m and n respectively. We investigate the Zariski closure Xt of the image Yt of the map HomK(V,W)→HomK(t⋀V,t⋀W), φ↦t⋀φ. In the case t=min(m,n), Yt=Xt is the cone over a Grassmannian, but for 1