Some Fano manifolds whose Hilbert polynomial is totally reducible over ℚ

Abstract

Let [Formula: see text] be any Fano manifold polarized by a positive multiple of its fundamental divisor [Formula: see text]. The polynomial defining the Hilbert curve of [Formula: see text] reduces to the Hilbert polynomial of [Formula: see text], hence it is totally reducible over [Formula: see text]; moreover, some of the linear factors appearing in the factorization have rational coefficients, e.g. if [Formula: see text] has index [Formula: see text]. It is natural to ask when the same happens for all linear factors. Here the total reducibility over [Formula: see text] 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.