{
  "id": "1010.4129",
  "version": "v2",
  "published": "2010-10-20T08:42:02.000Z",
  "updated": "2012-01-14T09:58:22.000Z",
  "title": "On the birational geometry of Fano 4-folds",
  "authors": [
    "Cinzia Casagrande"
  ],
  "comment": "43 pages. Final version, minor changes, to appear in Mathematische Annalen",
  "categories": [
    "math.AG"
  ],
  "abstract": "We study the birational geometry of a Fano 4-fold X from the point of view of Mori dream spaces; more precisely, we study rational contractions of X. Here a rational contraction is a rational map f: X-->Y, where Y is normal and projective, which factors as a finite sequence of flips, followed by a surjective morphism with connected fibers. Such f is called elementary if the difference of the Picard numbers of X and Y is 1. We first give a characterization of non-movable prime divisors in X, when X has Picard number at least 6; this is related to the study of birational and divisorial elementary rational contractions of X. Then we study the rational contractions of fiber type on X which are elementary or, more generally, quasi-elementary. The main result is that the Picard number of X is at most 11 if X has an elementary rational contraction of fiber type, and 18 if X has a quasi-elementary rational contraction of fiber type.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2012-01-14T09:58:22.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "birational geometry",
      "fiber type",
      "picard number",
      "divisorial elementary rational contractions",
      "mori dream spaces"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 43,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1010.4129C"
    }
  }
}