{
  "id": "math/0511659",
  "version": "v1",
  "published": "2005-11-27T17:17:29.000Z",
  "updated": "2005-11-27T17:17:29.000Z",
  "title": "On the Brill-Noether theory for K3 surfaces",
  "authors": [
    "Maxim Leyenson"
  ],
  "comment": "38 pages; xypic is used; 8 PostScript pictures",
  "categories": [
    "math.AG"
  ],
  "abstract": "Let (S,H) be a polarized K3 surface. We define Brill-Noether filtration on moduli spaces of vector bundles on S. Assume that (c_1(E),H) > 0 for a sheaf E in the moduli space. We give a formula for the expected dimension of the Brill-Noether subschemes. Following the classical theory for curves, we give a notion of Brill-Noether generic K3 surfaces. Studying correspondences between moduli spaces of sheaves of different ranks on S, we prove our main theorem: polarized K3 surface which is generic in sense of moduli is also generic in sense of Brill-Noether theory (here H is the positive generator of the Picard group of S). In case of algebraic curves such a theorem, proved by Griffiths and Harris and, independently, by Lazarsfeld, is sometimes called ``the strong theorem of the Brill-Noether theory''. We finish by considering a number of projective examples. In particular, we construct explicitly Brill-Noether special K3 surfaces of genus 5 and 6 and show the relation with the theory of Brill-Noether special curves.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2005-11-27T17:17:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14J10",
      "14J28",
      "14J60"
    ],
    "keywords": [
      "brill-noether theory",
      "moduli space",
      "polarized k3 surface",
      "explicitly brill-noether special k3 surfaces",
      "construct explicitly brill-noether special k3"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 38,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math.....11659L"
    }
  }
}