{
  "id": "1406.7404",
  "version": "v2",
  "published": "2014-06-28T14:19:31.000Z",
  "updated": "2018-09-06T12:46:40.000Z",
  "title": "A bound for Castelnuovo-Mumford regularity by double point divisors",
  "authors": [
    "Sijong Kwak",
    "Jinhyung Park"
  ],
  "comment": "23 pages. This paper has been largely rewritten after McCullough-Peeva's counterexamples to the Eisenbud-Goto regularity conjecture, which appeared in J. Amer. Math. Soc. in 2018. We also added new results on the regularity of smooth projective varieties of arbitrary dimension",
  "categories": [
    "math.AG",
    "math.AC"
  ],
  "abstract": "Let $X \\subseteq \\mathbb{P}^r$ be a non-degenerate smooth projective variety of dimension $n$, codimension $e$, and degree $d$ defined over an algebraically closed field of characteristic zero. In this paper, we first show that $\\text{reg} (\\mathcal{O}_X) \\leq d-e$, and classify the extremal and the next to extremal cases. Our result reduces the Eisenbud-Goto regularity conjecture for the smooth case to the problem finding a Castelnuovo-type bound for normality. It is worth noting that McCullough-Peeva recently constructed counterexamples to the regularity conjecture by showing that $\\text{reg} (\\mathcal{O}_X)$ is not even bounded above by any polynomial function of $d$ when $X$ is not smooth. For a normality bound in the smooth case, we establish that $\\text{reg}(X) \\leq n(d-2)+1$, which improves previous results obtained by Mumford, Bertram-Ein-Lazarsfeld, and Noma. Finally, by generalizing Mumford's method on double point divisors, we prove that $\\text{reg}(X) \\leq d-1+m$, where $m$ is an invariant arising from double point divisors associated to outer general projections. Using double point divisors associated to inner projection, we also obtain a slightly better bound for $\\text{reg}(X)$ under suitable assumptions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-06-28T14:19:31.000Z",
      "abstract": "Let $X \\subset \\mathbb{P}^r$ be a non-degenerate smooth projective variety of degree $d$ and codimension $e$. As motivated by the regularity conjecture due to Eisenbud and Goto, we first show that reg$ (\\mathcal{O}_X) \\leq d-e$, and we classify the extremal and the next extremal cases. Then, we generalize Mumford's method based on the geometric properties of double point divisors, and we prove reg$(X) \\leq d-1+m$, where $m$ is an invariant related to geometric sections of double point divisor from outer projection. By considering double point divisors from inner projection, we obtain a slightly better bound for Castelnuovo-Mumford regularity under suitable assumptions.",
      "comment": "17 pages, comments are welcome!",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2018-09-06T12:46:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14N05",
      "13D02",
      "14N25",
      "51N35"
    ],
    "keywords": [
      "castelnuovo-mumford regularity",
      "non-degenerate smooth projective variety",
      "extremal cases",
      "better bound",
      "mumfords method"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1406.7404K"
    }
  }
}