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

TL;DR

This paper investigates when the Hilbert polynomial of certain Fano manifolds factors completely over the rationals, focusing on cases with large index, low-dimensional toric Fano manifolds, and low coindex Fano bundles.

Contribution

It characterizes conditions under which the Hilbert polynomial of specific Fano manifolds is totally reducible over q, extending understanding of their algebraic and geometric properties.

Findings

Hilbert polynomial is totally reducible over q for Fano manifolds of large index.

Complete reducibility over q observed in low-dimensional toric Fano manifolds.

Fano bundles of low coindex also exhibit total reducibility of their Hilbert polynomial.

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 $\mathbb C$; moreover, some of the linear factors appearing in the factorization have rational coefficients, e.g. if $X$ has index $\geq 2$. It is natural to ask when the same happens for all linear factors. Here the total reducibility over $\mathbb Q$ 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.