{
  "id": "1402.3100",
  "version": "v1",
  "published": "2014-02-13T11:57:48.000Z",
  "updated": "2014-02-13T11:57:48.000Z",
  "title": "On Syzygies, degree, and geometric properties of projective schemes with property $\\textbf{N}_{3,p}$",
  "authors": [
    "Jeaman Ahn",
    "Sijong Kwak"
  ],
  "comment": "14 pages",
  "categories": [
    "math.AG",
    "math.AC"
  ],
  "abstract": "For an algebraic set $X$ (union of varieties) embedded in projective space, we say that $X$ satisfies property $\\textbf{N}_{d,p}$, $(d\\ge 2)$ if the $i$-th syzygies of the homogeneous coordinate ring are generated by elements of degree $< d+i$ for $0\\le i\\le p$ (see \\cite{EGHP2} for details). Much attention has been paid to linear syzygies of quadratic schemes $(d=2)$ and their geometric interpretations (cf. \\cite{AK},\\cite{EGHP1},\\cite{HK},\\cite{GL2},\\cite{KP}). However, not very much is actually known about the case satisfying property $\\textbf{N}_{3,p}$. In this paper, we give a sharp upper bound on the maximal length of a zero-dimensional linear section of $X$ in terms of graded Betti numbers (Theorem 1.2 (a)) when $X$ satisfies property $\\textbf{N}_{3,p}$. In particular, if $p$ is the codimension $e$ of $X$ then the degree of $X$ is less than or equal to $\\binom{e+2}{2}$, and equality holds if and only if $X$ is arithmetically Cohen-Maucalay with $3$-linear resolution (Theorem 1.2 (b)). This is a generalization of the results of Eisenbud et al. (\\cite{EGHP1,EGHP2}) to the case of $\\textbf{N}_{3,p}$, $(p\\leq e)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-02-13T11:57:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14N05",
      "13D02"
    ],
    "keywords": [
      "geometric properties",
      "projective schemes",
      "satisfies property",
      "sharp upper bound",
      "zero-dimensional linear section"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1402.3100A"
    }
  }
}