On the variety parametrizing 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 $\Split_{d}(\PP n)$, with the Grassmannian of $n-1$ dimensional projective subspaces of $\PP {n+d-1}$. We compute the dimension of some secant varieties to $\Split_{d}(\PP n)$ and find a counterexample to a conjecture that wanted its dimension related to the one of the secant variety to $\GG (n-1, n+d-1)$. Moreover by using an invariant embedding of the Veronse variety into the Pl\"ucker space, then we are able to compute the intersection of $\GG (n-1, n+d-1)$ with $\Split_{d}(\PP n)$, some of its secant variety, the tangential variety and the second osculating space to the Veronese variety.