{
  "id": "math/0601402",
  "version": "v1",
  "published": "2006-01-17T07:23:59.000Z",
  "updated": "2006-01-17T07:23:59.000Z",
  "title": "Normal generation and Clifford index",
  "authors": [
    "Youngook Choi",
    "Seonja Kim",
    "Young Rock Kim"
  ],
  "comment": "15pages, 2figures",
  "categories": [
    "math.AG"
  ],
  "abstract": "Let $C$ be a smooth curve of genus $g\\ge 4$ and Clifford index $c$. In this paper, we prove that if $C$ is neither hyperelliptic nor bielliptic with $g\\ge 2c+5$ and $\\mathcal M$ computes the Clifford index of $C$, then either $\\deg \\mathcal M\\le \\frac{3c}{2}+3$ or $|\\mathcal M|=|g^1_{c+2}+h^1_{c+2}|$ and $g=2c+5$. This strengthens the Coppens and Martens' theorem (\\cite{CM}, Corollary 3.2.5). Furthermore, for the latter case (1) $\\mathcal M$ is half-canonical unless $C$ is a $\\frac{c+2}{2}$-fold covering of an elliptic curve, (2) $\\mathcal M(F)$ fails to be normally generated with $\\cli(\\mathcal M(F))=c$, $h^1(\\mathcal M(F))=2$ for $F\\in g^1_{c+2}$. Such pairs $(C,\\mathcal M)$ can be found on a $K3$-surface whose Picard group is generated by a hyperplane section in $\\mathbb P^r$. For such a $(C, \\mathcal M)$ on a K3-surface, $\\mathcal M$ is normally generated while $\\mathcal M(F)$ fails to be normally generated with $\\cli(\\mathcal M)=\\cli(\\mathcal M(F))=c$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-01-17T07:23:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14H45",
      "14H10",
      "14C20",
      "14J10",
      "14J27",
      "14J28"
    ],
    "keywords": [
      "clifford index",
      "normal generation",
      "smooth curve",
      "elliptic curve"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......1402C"
    }
  }
}