{
  "id": "2011.06785",
  "version": "v1",
  "published": "2020-11-13T06:52:26.000Z",
  "updated": "2020-11-13T06:52:26.000Z",
  "title": "On the first non-trivial strand of syzygies of projective schemes and Condition ${\\mathrm ND}(l)$",
  "authors": [
    "Jeaman Ahn",
    "Kangjin Han",
    "Sijong Kwak"
  ],
  "comment": "19 pages, 3 figures",
  "categories": [
    "math.AG",
    "math.AC"
  ],
  "abstract": "Let $X\\subset\\mathbb{P}^{n+e}$ be any $n$-dimensional closed subscheme. In this paper, we are mainly interested in two notions related to syzygies: one is the property $\\mathbf{N}_{d,p}~(d\\ge 2, ~p\\geq 1)$, which means that $X$ is $d$-regular up to $p$-th step in the minimal free resolution and the other is a new notion $\\mathrm{ND}(l)$ 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 $l$. First, we introduce condition $\\mathrm{ND}(l)$ and consider examples and basic properties deduced from the notion. Next we prove sharp upper bounds on the graded Betti numbers of the first non-trivial 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 $\\mathbf{N}_{d,p}$, we characterize the resolution of $X$ to be $d$-linear arithemetically Cohen-Macaulay as having property $\\mathbf{N}_{d,e}$ and condition $\\mathrm{ND}(d-1)$ 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-Popescu to a general $d$-regularity.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-11-13T06:52:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14N05",
      "13D02",
      "51N35"
    ],
    "keywords": [
      "first non-trivial strand",
      "projective schemes",
      "general finite linear section",
      "syzygetic rigidity theorem",
      "sharp upper bounds"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}