{
  "id": "1103.3180",
  "version": "v3",
  "published": "2011-03-16T14:32:29.000Z",
  "updated": "2012-01-19T07:58:09.000Z",
  "title": "On Zariski's theorem in positive characteristic",
  "authors": [
    "Ilya Tyomkin"
  ],
  "comment": "19 pages. Several typos have been fixed, and a couple of examples and pictures have been added. To appear in JEMS",
  "categories": [
    "math.AG"
  ],
  "abstract": "In the current paper we show that the dimension of a family $V$ of irreducible reduced curves in a given ample linear system on a toric surface $S$ over an algebraically closed field is bounded from above by $-K_S.C+p_g(C)-1$, where $C$ denotes a general curve in the family. This result generalizes a famous theorem of Zariski to the case of positive characteristic. We also explore new phenomena that occur in positive characteristic: We show that the equality $\\dim(V)=-K_S.C+p_g(C)-1$ does not imply the nodality of $C$ even if $C$ belongs to the smooth locus of $S$, and construct reducible Severi varieties on weighted projective planes in positive characteristic, parameterizing irreducible reduced curves of given geometric genus in a given ample linear system.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2012-01-19T07:58:09.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "positive characteristic",
      "zariskis theorem",
      "ample linear system",
      "irreducible reduced curves",
      "construct reducible severi varieties"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1103.3180T"
    }
  }
}