Some Fano manifolds whose Hilbert polynomial is totally reducible over $\mathbb Q$

Abstract

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) boils down to being 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 Fano bundles of low coindex.