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