{
  "id": "1102.4550",
  "version": "v2",
  "published": "2011-02-22T16:52:13.000Z",
  "updated": "2011-07-15T16:09:24.000Z",
  "title": "The gonality theorem of Noether for hypersurfaces",
  "authors": [
    "Francesco Bastianelli",
    "Renza Cortini",
    "Pietro De Poi"
  ],
  "comment": "25 pages, final version",
  "journal": "J. Algebraic Geom. 23 (2014), 313-339",
  "categories": [
    "math.AG"
  ],
  "abstract": "It is well known since Noether that the gonality of a smooth plane curve of degree d>3 is d-1. Given a k-dimensional complex projective variety X, the most natural extension of gonality is probably the degree of irrationality, that is the minimum degree of a dominant rational map from X to $\\mathbb{P}^k$. In this paper we are aimed at extending the assertion on plane curves to smooth hypersurfaces in $\\mathbb{P}^n$ in terms of degree of irrationality. We prove that both surfaces in $\\mathbb{P}^3$ and threefolds in $\\mathbb{P}^4$ of sufficiently large degree d have degree of irrationality d-1, except for finitely many cases we classify, whose degree of irrationality is d-2. To this aim we use Mumford's technique of induced differentials and we shift the problem to study first order congruences of lines of $\\mathbb{P}^n$. In particular, we also slightly improve the description of such congruences in $\\mathbb{P}^4$ and we provide a bound on degree of irrationality of hypersurfaces of arbitrary dimension.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2011-07-15T16:09:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14E05",
      "14J70",
      "14N15"
    ],
    "keywords": [
      "gonality theorem",
      "irrationality",
      "study first order congruences",
      "k-dimensional complex projective variety",
      "dominant rational map"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1102.4550B"
    }
  }
}