{
  "id": "1901.02140",
  "version": "v1",
  "published": "2019-01-08T03:13:04.000Z",
  "updated": "2019-01-08T03:13:04.000Z",
  "title": "Rationality of Seshadri constants on general blow ups of $\\mathbb{P}^2$",
  "authors": [
    "Łucja Farnik",
    "Krishna Hanumanthu",
    "Jack Huizenga",
    "David Schmitz",
    "Tomasz Szemberg"
  ],
  "comment": "13 pages; comments welcome",
  "categories": [
    "math.AG"
  ],
  "abstract": "Let $X$ be a projective surface and let $L$ be an ample line bundle on $X$. The global Seshadri constant $\\varepsilon(L)$ of $L$ is defined as the infimum of Seshadri constants $\\varepsilon(L,x)$ as $x\\in X$ varies. It is an interesting question to ask if $\\varepsilon(L)$ is a rational number for any pair $(X, L)$. We study this question when $X$ is a blow up of $\\mathbb{P}^2$ at $r \\ge 0$ very general points and $L$ is an ample line bundle on $X$. For each $r$ we define a $\\textit{submaximality threshold}$ which governs the rationality or irrationality of $\\varepsilon(L)$. We state a conjecture which strengthens the SHGH Conjecture and assuming that this conjecture is true we determine the submaximality threshold.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-01-08T03:13:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14C20",
      "14H50",
      "14J26"
    ],
    "keywords": [
      "general blow ups",
      "ample line bundle",
      "rationality",
      "global seshadri constant",
      "submaximality threshold"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}