{
  "id": "1105.3449",
  "version": "v1",
  "published": "2011-05-17T18:55:56.000Z",
  "updated": "2011-05-17T18:55:56.000Z",
  "title": "Big $q$-ample Line Bundles",
  "authors": [
    "Morgan V Brown"
  ],
  "comment": "9 pages, 2 pdf figures",
  "doi": "10.1112/S0010437X11007457",
  "categories": [
    "math.AG"
  ],
  "abstract": "A recent paper of Totaro develops a theory of $q$-ample bundles in characteristic 0. Specifically, a line bundle $L$ on $X$ is $q$-ample if for every coherent sheaf $\\mathcal{F}$ on $X$, there exists an integer $m_0$ such that $m\\geq m_0$ implies $H^i(X,\\mathcal{F}\\otimes \\mathcal{O}(mL))=0$ for $i>q$. We show that a line bundle $L$ on a complex projective scheme $X$ is $q$-ample if and only if the restriction of $L$ to its augmented base locus is $q$-ample. In particular, when $X$ is a variety and $L$ is big but fails to be $q$-ample, then there exists a codimension 1 subscheme $D$ of $X$ such that the restriction of $L$ to $D$ is not $q$-ample.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-05-17T18:55:56.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14C20"
    ],
    "keywords": [
      "ample line bundles",
      "coherent sheaf",
      "ample bundles",
      "complex projective scheme",
      "restriction"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1105.3449B"
    }
  }
}