Linear subvarieties of Cayley-Chow varieties

We apply the results of our previous paper [Degrees ...] to the projective geometry of the Chow variety which parametrizes cycles of dimension n and degree k in a projective space P^r. We show that, for k>1, its linear subvarieties are the varieties parametrizing linear systems of hypersurfaces in some P^{n+1} contained in P^r, plus a possible fixed cycle not supported in P^{n+1}, or the varieties parametrizing linear families of linear subspaces plus a fixed cycle of degree k-1. This extends the known description of linear subvarieties of a Grassmannian, in the case k=1.