{
  "id": "2104.11443",
  "version": "v1",
  "published": "2021-04-23T07:15:31.000Z",
  "updated": "2021-04-23T07:15:31.000Z",
  "title": "Flops and Mordell-Weil group of Elliptic Threefolds with (4,6,12)-singular fibers",
  "authors": [
    "David Wen"
  ],
  "comment": "16 pages; Comments welcome",
  "categories": [
    "math.AG",
    "hep-th"
  ],
  "abstract": "Let $f: W \\rightarrow T$ be an elliptic threefold that is a Weierstrass model, which is locally defined by $y^2 = x^3 + fx + g$ over $T$, with a singular fiber such that $(f,g,4f^2 + 27g^3)$ vanishes of order $(4,6,12)$ over an isolated point over $T$. Such a fiber can be resolved to a terminal model, $Y$, containing a rational elliptic surface, $S$, where some sections of $S$ are flopping curves on $Y$. As a consequence of this arithmetic and geometric connection, we are able to bound of the free rank of Mordell-Weil group of $W \\rightarrow T$ below by the free rank of the Mordell-Weil group of $S$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-04-23T07:15:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14E07",
      "14J27"
    ],
    "keywords": [
      "mordell-weil group",
      "elliptic threefold",
      "free rank",
      "rational elliptic surface",
      "singular fiber"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}