{
  "id": "2210.11659",
  "version": "v1",
  "published": "2022-10-21T01:06:09.000Z",
  "updated": "2022-10-21T01:06:09.000Z",
  "title": "Torsion points and concurrent exceptional curves on del Pezzo surfaces of degree one",
  "authors": [
    "Julie Desjardins",
    "Rosa Winter"
  ],
  "comment": "14 pages",
  "categories": [
    "math.AG",
    "math.NT"
  ],
  "abstract": "The blow-up of the anticanonical base point on a del Pezzo surface $X$ of degree 1 gives rise to a rational elliptic surface $\\mathscr{E}$ with only irreducible fibers. The sections of minimal height of $\\mathscr{E}$ are in correspondence with the $240$ exceptional curves on $X$. A natural question arises when studying the configuration of these curves: if a point on $X$ is contained in 'many' exceptional curves, it is torsion on its fiber on $\\mathscr{E}$? In 2005, Kuwata proved for the analogous question on del Pezzo surfaces of degree $2$, where there are 56 exceptional curves, that if 'many' equals $4$ or more, the answer is yes. In this paper, we prove that for del Pezzo surfaces of degree 1, the answer is yes if 'many' equals $9$ or more. Moreover, we give counterexamples where a \\textsl{non}-torsion point lies in the intersection of $7$ exceptional curves. We give partial results for the still open case of 8 intersecting exceptional curves.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-10-21T01:06:09.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "del pezzo surface",
      "concurrent exceptional curves",
      "torsion points",
      "rational elliptic surface",
      "natural question arises"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}