{
  "id": "math/9609212",
  "version": "v1",
  "published": "1996-09-04T00:00:00.000Z",
  "updated": "1996-09-04T00:00:00.000Z",
  "title": "The space of rational maps on P^1",
  "authors": [
    "Joseph H. Silverman"
  ],
  "journal": "Duke Math. J. 94 (1998), 41--77",
  "categories": [
    "math.DS",
    "math.NT"
  ],
  "abstract": "The set of morphisms $\\f:\\PP^1\\to\\PP^1$ of degree $d$ is parametrized by an affine open subset $\\Rat_d$ of $\\PP^{2d+1}$. We consider the action of~$\\SL_2$ on $\\Rat_d$ induced by the {\\it conjugation action\\/} of $\\SL_2$ on rational maps; that is, $f\\in\\SL_2$ acts on~$\\f$ via $\\f^f=f^{-1}\\circ\\f\\circ f$. The quotient space $\\M_d=\\Rat_d/\\SL_2$ arises very naturally in the study of discrete dynamical systems on~$\\PP^1$. We prove that~$\\M_d$ exists as an affine integral scheme over~$\\ZZ$, that $\\M_2$ is isomorphic to~$\\AA^2_\\ZZ$, and that the natural completion of~$\\M_2$ obtained using geometric invariant theory is isomorphic to~$\\PP^2_\\ZZ$. These results, which generalize results of Milnor over~$\\CC$, should be useful for studying the arithmetic properties of dynamical systems.",
  "revisions": [
    {
      "version": "v1",
      "updated": "1996-09-04T00:00:00.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "rational maps",
      "geometric invariant theory",
      "affine integral scheme",
      "affine open subset",
      "quotient space"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "1996math......9212S"
    }
  }
}