{
  "id": "0905.3239",
  "version": "v4",
  "published": "2009-05-20T08:16:21.000Z",
  "updated": "2011-12-20T20:39:22.000Z",
  "title": "On the Picard number of divisors in Fano manifolds",
  "authors": [
    "C. Casagrande"
  ],
  "comment": "Final version, to appear in the Annales Scientifiques de l'Ecole Normale Superieure",
  "categories": [
    "math.AG"
  ],
  "abstract": "Let X be a complex Fano manifold of arbitrary dimension, and D a prime divisor in X. We consider the image H of N_1(D) in N_1(X) under the natural push-forward of 1-cycles. We show that the codimension c of H in N_1(X) is at most 8. Moreover if c>2, then either X=SxY where S is a Del Pezzo surface, or c=3 and X has a flat fibration in Del Pezzo surfaces onto a Fano manifold Y, such that the difference of the Picard numbers of X and Y is 4. We give applications to Fano 4-folds, to Fano varieties with pseudo-index >1, and to surjective morphisms whose source is Fano, having some high-dimensional fibers or low-dimensional target.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2011-12-20T20:39:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14J45"
    ],
    "keywords": [
      "picard number",
      "del pezzo surface",
      "complex fano manifold",
      "low-dimensional target",
      "arbitrary dimension"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0905.3239C"
    }
  }
}