Some Fano manifolds whose Hilbert polynomial is totally reducible over ℚ

Let (X, L) be any Fano manifold polarized by a positive multiple of its fundamental divisor H. The polynomial defining the Hilbert curve of (X, L) reduces to the Hilbert polynomial of (X, H), hence it is totally reducible over C; moreover, some of the linear factors appearing in the factorization have rational coefficients, e.g. if X has index >= 2. It is natural to ask when the same happens for all linear factors. Here the total reducibility over Q of the Hilbert polynomial is investigated for three special kinds of Fano manifolds: Fano manifolds of large index, toric Fano manifolds of low dimension, and projectivized Fano bundles of low coindex.