{
  "id": "1408.2233",
  "version": "v2",
  "published": "2014-08-10T14:48:19.000Z",
  "updated": "2015-09-20T10:34:28.000Z",
  "title": "Rationality problem of conic bundles",
  "authors": [
    "Aiichi Yamasaki"
  ],
  "comment": "incorporates all of the content of arXiv:1308.0909",
  "categories": [
    "math.AG"
  ],
  "abstract": "Let $k$ be a field with char $k \\not= 2$, $X$ be an affine surface defined by the equation $z^2=P(x)y^2+Q(x)$ where $P(x), Q(x) \\in k[x]$ are separable polynomials. We will investigate the rationality problem of $X$ in terms of the polynomials $P(x)$ and $Q(x)$. The necessary and sufficient condition is $s \\leq 3$ with minor exceptions, where $s=s_1+s_2+s_3+s_4$, $s_1$ (resp. $s_2$, resp. $s_3$) being the number of $c \\in \\overline{k}$ such that $P(c)=0$ and $Q(c) \\not\\in k(c)^2$ (resp. $Q(c)=0$ and $P(c) \\not\\in k(c)^2$, resp. $P(c)=Q(c)=0$ and $-\\frac{Q}{P}(c) \\not\\in k(c)^2$). $s_4=0$ or $1$ according to the behavior at $x=\\infty$. $X$ is a conic bundle over $\\mathbb{P}_k^1$, whose rationality was studied by Iskovskikh. Iskovskikh formulated his results in geometric language. This paper aims to give an algebraic counterpart.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-08-10T14:48:19.000Z",
      "abstract": "Let $k$ be a field with char $k \\not= 2$, $X$ be an affine surface defined by the equation $z^2=P(x)Y^2+Q(x)$ where $P(x), Q(x) \\in k[x]$ are separable polynomials. We will investigate the rationality problem of $X$ in terms of the polynomials $P(x)$ and $Q(x)$. The necessary and sufficient condition is $s \\leq 3$ with minor exceptions, where $s=s_1+s_2+s_3+s_4$, $s_1$ (resp. $s_2$, resp. $s_3$) being the number of $c \\in \\overline{k}$ such that $P(c)=0$ and $Q(c) \\not\\in k(c)^2$ (resp. $Q(c)=0$ and $P(c) \\not\\in k(c)^2$, resp. $P(c)=Q(c)=0$ and $-\\frac{Q}{P}(c) \\not\\in k(c)^2$). $s_4=0$ or $1$ according to the behavior at $x=\\infty$. $X$ is a conic bundle over $\\mathbb{P}_k^1$, whose rationality was studied by Iskovskikh. Iskovskikh formulated his results in geometric language. This paper aims to give an algebraic counterpart.",
      "comment": "text overlap with arXiv:1308.0909",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-09-20T10:34:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14E08",
      "14E05",
      "12G05"
    ],
    "keywords": [
      "rationality problem",
      "conic bundle",
      "polynomials",
      "sufficient condition",
      "algebraic counterpart"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1408.2233Y"
    }
  }
}