{
  "id": "math/0505266",
  "version": "v4",
  "published": "2005-05-12T15:41:06.000Z",
  "updated": "2006-09-19T15:04:04.000Z",
  "title": "Singular curves on a K3 surface and linear series on their normalizations",
  "authors": [
    "Flaminio Flamini",
    "Andreas Leopold Knutsen",
    "Gianluca Pacienza"
  ],
  "comment": "20 pages, remarks of the referee are added, to be published on International Journal of Mathematics",
  "categories": [
    "math.AG"
  ],
  "abstract": "In this paper, we study the Brill-Noether theory of the normalizations of singular, irreducible curves on a $K3$ surface. We introduce a {\\em singular} Brill-Noether number $\\rho_{sing}$ and show that if the Picard group of the K3 surface is ${\\mathbb Z} [L]$, there are no $g^r_d$'s on the normalizations of irreducible curves in $|L|$, provided that $\\rho_{sing} <0$. We give examples showing the sharpness of this result. We then focus on the case of {\\em hyperelliptic normalizations}, and classify linear systems $|L|$ containing irreducible nodal curves with hyperelliptic normalizations, for $\\rho_{sing}<0$, without any assumption on its Picard group.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2006-09-19T15:04:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14H10",
      "14H51",
      "14J28",
      "14J60"
    ],
    "keywords": [
      "k3 surface",
      "linear series",
      "singular curves",
      "picard group",
      "hyperelliptic normalizations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math......5266F"
    }
  }
}