Linearity of free resolutions of monomial ideals

Dao, Hailong; Eisenbud, David

doi:10.1007/s40687-022-00330-6

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2023

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Cited by 3 publications

(1 citation statement)

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“…Theorem 1.1 helped launch the modern study of the geometry of syzygies. It led to numerous generalizations involving embeddings of surfaces [AKL19, BH13, GP96, GP99, KL19, NY22], smooth higher dimensional varieties [EGHP05, EL93, GP98, HSS06, HT13], abelian varieties [Chi19,Par00,PP03,PP04], and Calabi-Yau varieties [Niu19]; see also [DE22,ENP20]. Theorem 1.1 also led to Green's Conjecture, which proposes a relationship between the Clifford index of a non-hyperelliptic curve over C and the linearity of the free resolution of its coordinate ring with respect to its canonical embedding [Gre84b].…”

Section: Introductionmentioning

confidence: 99%

Linear syzygies of curves in weighted projective space

Brown¹,

Erman²

2023

Preprint

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Section: Introductionmentioning

confidence: 99%

Linear syzygies of curves in weighted projective space

Brown¹,

Erman²

2023

Preprint

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Burch Index, Summands of Syzygies and Linearity in Resolutions

Dao

Eisenbud

2023

Bull. Iran. Math. Soc.

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Bounds for Betti numbers and the degrees of the generators of syzygies in graded resolutions

2023

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The main goal of this paper is to size up the minimal graded free resolution of a homogeneous ideal in terms of its generating degrees, by itself an ambitious objective. As understood, sizing up means looking closely at the two available parameters: the shifts and the Betti numbers. Since, in general, bounds for the shifts can behave quite steeply, we filter the difficulty by the subadditivity of the syzygies. The method we applied is hopefully new and sheds additional light on the structure of the minimal free resolution. For the Betti numbers, we apply the Boij–Söderberg techniques in order to get polynomial upper bounds for them. It is expected that the landscape of hypersurface singularities already contains most of the difficult corners of the arbitrary case. In this regard, we treat some facets of this case, including an improvement of one of the basic results of a recent work by Busé–Dimca–Schenck–Sticlaru.

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Linearity of free resolutions of monomial ideals

Cited by 3 publications

References 15 publications

Linear syzygies of curves in weighted projective space

Linear syzygies of curves in weighted projective space

Burch Index, Summands of Syzygies and Linearity in Resolutions

Bounds for Betti numbers and the degrees of the generators of syzygies in graded resolutions

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