{
  "id": "1408.5655",
  "version": "v1",
  "published": "2014-08-25T03:13:13.000Z",
  "updated": "2014-08-25T03:13:13.000Z",
  "title": "Automorphism loci for the moduli space of rational maps",
  "authors": [
    "Nikita Miasnikov",
    "Brian Stout",
    "Phillip Williams"
  ],
  "comment": "29 pages",
  "categories": [
    "math.DS"
  ],
  "abstract": "Let $k$ be an algebraically closed field of characteristic $0$ and $\\mathcal{M}_d$ the moduli space of rational maps on $\\mathbb{P}^1$ of degree $d$ over $k$. This paper describes the automorphism loci $A\\subset \\mathrm{Rat}_d$ and $\\mathcal{A}\\subset \\mathcal{M}_d$ and the singular locus $\\mathcal{S}\\subset\\mathcal{M}_d$. In particular, we determine which groups occur as subgroups of the automorphism group of some $[\\phi]\\in\\mathcal{M}_d$ for a given $d$ and calculate the dimension of the locus. Next, we prove an analogous theorem to the Rauch-Popp-Oort characterization of singular points on the moduli scheme for curves. The results concerning these distinguished loci are used to compute the Picard and class groups of $\\mathcal{M}_d, \\mathcal{M}^s_d,$ and $\\mathcal{M}^{ss}_d$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-08-25T03:13:13.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "moduli space",
      "rational maps",
      "automorphism loci",
      "rauch-popp-oort characterization",
      "class groups"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1408.5655M"
    }
  }
}