{
  "id": "1705.00901",
  "version": "v1",
  "published": "2017-05-02T10:50:01.000Z",
  "updated": "2017-05-02T10:50:01.000Z",
  "title": "Plane model-fields of definition, fields of definition, the field of moduli of smooth plane curves",
  "authors": [
    "Eslam Badr",
    "Francesc Bars"
  ],
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "Given a smooth plane curve $\\overline{C}$ of genus $g\\geq 3$ over an algebraically closed field $\\overline{k}$, a field $L\\subseteq\\overline{k}$ is said to be a \\emph{plane model-field of definition for $\\overline{C}$} if $L$ is a field of definition for $\\overline{C}$, i.e. $\\exists$ a smooth curve $C'$ defined over $L$ where $C'\\times_L\\overline{k}\\cong \\overline{C}$, and such that $C'$ is $L$-isomorphic to a non-singular plane model $F(X,Y,Z)=0$ in $\\mathbb{P}^2_{L}$. {In this short note, we construct a smooth plane curve $\\overline{C}$ over $\\overline{\\mathbb{Q}}$, such that the field of moduli of $\\overline{C}$ is not a field of definition for $\\overline{C}$, and also fields of definition do not coincide with plane model-fields of definition for $\\overline{C}$.} As far as we know, this is the first example in the literature with the above property, since this phenomenon does not occur for hyperelliptic curves, replacing plane model-fields of definition with the so-called hyperelliptic model-fields of definition.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-05-02T10:50:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14H45",
      "14G27",
      "14H37",
      "11G30"
    ],
    "keywords": [
      "smooth plane curve",
      "definition",
      "non-singular plane model",
      "hyperelliptic model-fields",
      "smooth curve"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}