{
  "id": "1709.05698",
  "version": "v1",
  "published": "2017-09-17T17:51:32.000Z",
  "updated": "2017-09-17T17:51:32.000Z",
  "title": "The rationality problem for forms of $\\overline{M_{0, n}}$",
  "authors": [
    "Mathieu Florence",
    "Zinovy Reichstein"
  ],
  "comment": "12 pages",
  "categories": [
    "math.AG",
    "math.GR",
    "math.NT"
  ],
  "abstract": "Let $X$ be a del Pezzo surface of degree $5$ defined over a field $F$. A theorem of Yu. I. Manin and P. Swinnerton-Dyer asserts that every Del Pezzo surface of degree $5$ is rational. In this paper we generalize this result as follows. Recall that del Pezzo surfaces of degree $5$ over a field $F$ are precisely the twisted $F$-forms of the moduli space $\\overline{M_{0, 5}}$ of stable curves of genus $0$ with $5$ marked points. Suppose $n \\geq 5$ is an integer, and $F$ is an infinite field of characteristic $\\neq 2$. It is easy to see that every twisted $F$-form of $\\overline{M_{0, n}}$ is unirational over $F$. We show that (a) if $n$ is odd, then every twisted $F$-form of $\\overline{M_{0, n}}$ is rational over $F$. (b) If $n$ is even, there exists a field extension $F/k$ and a twisted $F$-form $X$ of $\\overline{M_{0, n}}$ such that $X$ is not retract rational over $F$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-09-17T17:51:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14E08",
      "14H10",
      "20G15",
      "16K50"
    ],
    "keywords": [
      "del pezzo surface",
      "rationality problem",
      "infinite field",
      "moduli space",
      "swinnerton-dyer asserts"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}