{
  "id": "2210.04789",
  "version": "v2",
  "published": "2022-10-10T15:45:48.000Z",
  "updated": "2022-10-12T15:28:08.000Z",
  "title": "Fields of moduli and the arithmetic of tame quotient singularities",
  "authors": [
    "Giulio Bresciani",
    "Angelo Vistoli"
  ],
  "categories": [
    "math.AG",
    "math.NT"
  ],
  "abstract": "Given a perfect field $k$ with algebraic closure $\\bar{k}$ and a variety $X$ over $\\bar{k}$, the field of moduli of $X$ is the subfield of $\\bar{k}$ of elements fixed by field automorphisms $\\gamma\\in\\operatorname{Gal}(\\bar{k}/k)$ such that the twist $X_{\\gamma}$ is isomorphic to $X$. The field of moduli is contained in all subextensions $k\\subset k'\\subset\\bar{k}$ such that $X$ descends to $k'$. In this paper we extend the formalism, and define the field of moduli when $k$ is not perfect. D\\`ebes and Emsalem identified a condition that ensures that a smooth curve is defined over its field of moduli, and proved that a smooth curve with a marked point is always defined over its field of moduli. Our main theorem is a generalization of these results that applies to higher dimensional varieties, and to varieties with additional structures. In order to apply this, we study the problem of when a rational point of a variety with quotient singularities lifts to a resolution. As an application we give conditions on the automorphism group of a variety $X$ with a smooth marked point $p$ that ensure that the pair $(X,p)$ is defined over its field of moduli.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-10-12T15:28:08.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "tame quotient singularities",
      "arithmetic",
      "smooth curve",
      "quotient singularities lifts",
      "higher dimensional varieties"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}