{
  "id": "1407.7845",
  "version": "v3",
  "published": "2014-07-29T19:47:20.000Z",
  "updated": "2014-08-14T22:29:00.000Z",
  "title": "Rational curves on elliptic surfaces",
  "authors": [
    "Douglas Ulmer"
  ],
  "comment": "15 pages. v2: Added a reference and corrected a quote. v3: Added another reference",
  "categories": [
    "math.AG"
  ],
  "abstract": "We prove that a very general elliptic surface $\\mathcal{E}\\to\\mathbb{P}^1$ over the complex numbers with a section and with geometric genus $p_g\\ge2$ contains no rational curves other than the section and components of singular fibers. Equivalently, if $E/\\mathbb{C}(t)$ is a very general elliptic curve of height $d\\ge3$ and if $L$ is a finite extension of $\\mathbb{C}(t)$ with $L\\cong\\mathbb{C}(u)$, then the Mordell-Weil group $E(L)=0$.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2014-08-14T22:29:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14J27",
      "14G99",
      "11G99"
    ],
    "keywords": [
      "rational curves",
      "general elliptic surface",
      "general elliptic curve",
      "geometric genus",
      "singular fibers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1407.7845U"
    }
  }
}