On the variety parameterizing completely decomposable polynomials

The purpose of this paper is to relate the variety parameterizing completely decomposable homogeneous polynomials of degree d in n+1 variables on an algebraically closed field, called , with the Grassmannian of (n−1)-dimensional projective subspaces of Pn+d−1. We compute the dimension of some secant varieties to . Moreover by using an invariant embedding of the Veronese variety into the Plücker space, we are able to compute the intersection of G(n−1,n+d−1) with , some of its secant varieties, the tangential variety and the second osculating space to the Veronese variety.