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