{
  "id": "1605.05018",
  "version": "v1",
  "published": "2016-05-17T05:06:35.000Z",
  "updated": "2016-05-17T05:06:35.000Z",
  "title": "Geometry of webs of algebraic curves",
  "authors": [
    "Jun-Muk Hwang"
  ],
  "comment": "35 pages, to appear in Duke Math. J",
  "categories": [
    "math.AG",
    "math.CV",
    "math.DG"
  ],
  "abstract": "A family of algebraic curves covering a projective variety $X$ is called a web of curves on $X$ if it has only finitely many members through a general point of $X$. A web of curves on $X$ induces a web-structure, in the sense of local differential geometry, in a neighborhood of a general point of $X$. We study how the local differential geometry of the web-structure affects the global algebraic geometry of $X$. Under two geometric assumptions on the web-structure, the pairwise non-integrability condition and the bracket-generating condition, we prove that the local differential geometry determines the global algebraic geometry of $X$, up to generically finite algebraic correspondences. The two geometric assumptions are satisfied, for example, when $X \\subset {\\bf P}^N$ is a Fano submanifold of Picard number 1, and the family of lines covering $X$ becomes a web. In this special case, we have a stronger result that the local differential geometry of the web-structure determines $X$ up to biregular equivalences. As an application, we show that if $X, X' \\subset {\\bf P}^N, \\dim X' \\geq 3,$ are two such Fano manifolds of Picard number 1, then any surjective morphism $f: X \\to X'$ is an isomorphism.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-05-17T05:06:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14M22",
      "32D15",
      "14J45",
      "32H04",
      "53A60"
    ],
    "keywords": [
      "algebraic curves",
      "global algebraic geometry",
      "web-structure",
      "picard number",
      "geometric assumptions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}