{
  "id": "math/9805140",
  "version": "v2",
  "published": "1998-05-29T09:46:03.000Z",
  "updated": "2001-06-07T19:54:58.000Z",
  "title": "Smooth curves on projective K3 surfaces",
  "authors": [
    "Andreas Leopold Knutsen"
  ],
  "comment": "12 pages, to appear in Math. Scand. Mistake in earlier version of Thm 1.1 corrected and its proof is considerably simplified (removed the now redundant Sections 4 and 5 of the previous version). Added Rem. 1.2 and Prop. 1.3",
  "categories": [
    "math.AG"
  ],
  "abstract": "In this paper we give for all $n \\geq 2$, d>0, $g \\geq 0$ necessary and sufficient conditions for the existence of a pair (X,C), where X is a K3 surface of degree 2n in $\\matbf{P}^{n+1}$ and C is a smooth (reduced and irreducible) curve of degree d and genus g on X. The surfaces constructed have Picard group of minimal rank possible (being either 1 or 2), and in each case we specify a set of generators. For $n \\geq 4$ we also determine when X can be chosen to be an intersection of quadrics (in all other cases X has to be an intersection of both quadrics and cubics). Finally, we give necessary and sufficient conditions for $\\O_C (k)$ to be non-special, for any integer $k \\geq 1$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2001-06-07T19:54:58.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "projective k3 surfaces",
      "smooth curves",
      "sufficient conditions",
      "picard group",
      "degree 2n"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "1998math......5140L"
    }
  }
}