{
  "id": "2007.05818",
  "version": "v1",
  "published": "2020-07-11T17:47:36.000Z",
  "updated": "2020-07-11T17:47:36.000Z",
  "title": "On a rationality problem for fields of cross-ratios II",
  "authors": [
    "Tran-Trung Nghiem",
    "Zinovy Reichstein"
  ],
  "comment": "9 pages",
  "categories": [
    "math.AG",
    "math.AC",
    "math.GR"
  ],
  "abstract": "Let $k$ be a field, $x_1, \\dots, x_n$ be independent variables and $L_n = k(x_1, \\dots, x_n)$. The symmetric group $\\Sigma_n$ acts on $L_n$ by permuting the variables, and the projective linear group $\\text{PGL}_2$ acts by \\[ \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} \\colon x_i \\mapsto \\frac{a x_i + b}{c x_i + d} \\] for each $i = 1, \\ldots, n$. The fixed field $L_n^{\\text{PGL}_2}$ is called \"the field of cross-ratios\". Given a subgroup $S \\subset \\Sigma_n$, H. Tsunogai asked whether $L_n^S$ rational over $K_n^S$. When $n \\geqslant 5$ the second author has shown that $L_n^S$ is rational over $K_n^S$ if and only if $S$ has an orbit of odd order in $\\{ 1, \\dots, n \\}$. In this paper we answer Tsunogai's question for $n \\leqslant 4$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-07-11T17:47:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14E08"
    ],
    "keywords": [
      "rationality problem",
      "cross-ratios",
      "answer tsunogais question",
      "symmetric group",
      "independent variables"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}