{
  "id": "1201.4807",
  "version": "v3",
  "published": "2012-01-23T18:46:56.000Z",
  "updated": "2015-11-14T01:02:29.000Z",
  "title": "Torus Quotients as Global Quotients by Finite Groups",
  "authors": [
    "Anton Geraschenko",
    "Matthew Satriano"
  ],
  "comment": "Updated to match published version. Note: the appendix (including Theorem 6.1) has been moved to arXiv:1503.05478",
  "categories": [
    "math.AG"
  ],
  "abstract": "This article is motivated by the following local-to-global question: is every variety with tame quotient singularities globally the quotient of a smooth variety by a finite group? We show that this question has a positive answer for all quasi-projective varieties which are expressible as a quotient of a smooth variety by a split torus (e.g. simplicial toric varieties). Although simplicial toric varieties are rarely toric quotients of smooth varieties by finite groups, we give an explicit procedure for constructing the quotient structure using toric techniques. This result follow from a characterization of varieties which are expressible as the quotient of a smooth variety by a split torus. As an additional application of this characterization, we show that a variety with abelian quotient singularities may fail to be a quotient of a smooth variety by a finite abelian group. Concretely, we show that $\\mathbb{P}^2/A_5$ is not expressible as a quotient of a smooth variety by a finite abelian group.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2012-05-15T22:48:32.000Z",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2015-11-14T01:02:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14L30",
      "14D23",
      "14M25"
    ],
    "keywords": [
      "finite group",
      "smooth variety",
      "torus quotients",
      "global quotients",
      "simplicial toric varieties"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1201.4807G"
    }
  }
}