Examples Violating Golyshev’s Canonical Strip Hypotheses

Belmans, Pieter; Galkin, Sergey; Mukhopadhyay, Swarnava

doi:10.1080/10586458.2019.1602571

Cited by 2 publications

(3 citation statements)

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“…A large literature is devoted to it and some of its variants, starting with Golyshev's paper [9] (e.g. see [17], [10], and [3], where examples disproving the hypothesis for n ≥ 7 are provided). Another natural question, which is what we deal with in this paper, is about the rationality of all roots, as it happens in the case of P n .…”

Section: Introductionmentioning

confidence: 99%

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Some Fano manifolds whose Hilbert polynomial is totally reducible over ℚ

Lanteri

Tironi²

2023

Int. J. Math.

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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.

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Section: Introductionmentioning

confidence: 99%

“…3 + 117z 2 + 157z + 60), which is totally reducible over R but not over Q. Furthermore, for 4 ≤ m ≤ 150, one can use the following Magma Program[6]:…”

mentioning

confidence: 99%

Some Fano manifolds whose Hilbert polynomial is totally reducible over ℚ

Lanteri

Tironi²

2023

Int. J. Math.

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show abstract

“…A large literature is devoted to it and some of its variants, starting with Golyshev's paper [9] (e.g. see [17], [10], [3]). Another natural question, which is what we deal with in this paper, is about the rationality of all roots, as it happens in the case of P n .…”

Section: Introductionmentioning

confidence: 99%

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

Lanteri¹,

Tironi²

2022

Preprint

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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.

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Examples Violating Golyshev’s Canonical Strip Hypotheses

Cited by 2 publications

References 7 publications

Some Fano manifolds whose Hilbert polynomial is totally reducible over ℚ

Some Fano manifolds whose Hilbert polynomial is totally reducible over ℚ

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

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