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On the first nontrivial strand of syzygies of projective schemes and condition $\mathrm{ND}(\ell)$

Jeaman Ahn, Kangjin Han and Sijong Kwak

Vol. 17 (2023), No. 8, 1359–1380
Abstract

Let X n+e be any n-dimensional closed subscheme. We are mainly interested in two notions related to syzygies: one is the property Nd,p(d 2,p 1), which means that X is d-regular up to p-th step in the minimal free resolution and the other is a new notion ND () which generalizes the classical “being nondegenerate” to the condition that requires a general finite linear section not to be contained in any hypersurface of degree .

First, we introduce condition ND () and consider examples and basic properties deduced from the notion. Next we prove sharp upper bounds on the graded Betti numbers of the first nontrivial strand of syzygies, which generalize results in the quadratic case to higher degree case, and provide characterizations for the extremal cases. Further, after regarding some consequences of property Nd,p, we characterize the resolution of X to be d-linear arithmetically Cohen–Macaulay as having property Nd,e and condition ND (d 1) at the same time. From this result, we obtain a syzygetic rigidity theorem which suggests a natural generalization of syzygetic rigidity on 2-regularity due to Eisenbud, Green, Hulek and Popescu to a general d-regularity.

Keywords
graded Betti numbers, higher linear syzygies, condition ND(l), property $\mathbf{N}_{d,p}$, arithmetically Cohen–Macaulay, Castelnuovo–Mumford regularity
Mathematical Subject Classification
Primary: 13D02, 14N05
Secondary: 51N35
Milestones
Received: 26 November 2020
Revised: 8 July 2022
Accepted: 6 September 2022
Published: 29 August 2023
Authors
Jeaman Ahn
Department of Mathematics Education
Kongju National University
Kongju
South Korea
Kangjin Han
School of Undergraduate Studies
Daegu-Gyeongbuk Institute of Science & Technology (DGIST)
Daegu
South Korea
Sijong Kwak
Department of Mathematical Sciences
KAIST
Daejeon
South Korea

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