{
  "id": "1407.2284",
  "version": "v2",
  "published": "2014-07-08T22:15:37.000Z",
  "updated": "2015-11-08T18:18:51.000Z",
  "title": "On the rigidity of moduli of curves in arbitrary characteristic",
  "authors": [
    "Barbara Fantechi",
    "Alex Massarenti"
  ],
  "comment": "Streamlined version. We added a study of the deformations of the coarse moduli scheme \\bar{M}_{g,n} in characteristic zero (a question left open by P. Hacking in arXiv:math/0509567): via its description as a toric surface we show that \\bar{M}_{1,2} has a 6-dimensional family of infinitesimal deformations and is smoothable, while \\bar{M}_{g,n} is rigid for g+n>4",
  "categories": [
    "math.AG"
  ],
  "abstract": "The stack $\\overline{\\mathcal{M}}_{g,n}$ of stable curves and its coarse moduli space $\\overline{M}_{g,n}$ are defined over $\\mathbb{Z}$, and therefore over any field. Over an algebraically closed field of characteristic zero, Hacking showed that $\\overline{\\mathcal{M}}_{g,n}$ is rigid (a conjecture of Kapranov). Bruno and Mella for $g=0$, and the second author for $g\\geq 1$ showed that its automorphism group is the symmetric group $S_n$, permuting marked points unless $(g,n)\\in\\{(0,4),(1,1),(1,2)\\}$. The methods used in the papers above do not extend to positive characteristic. We show that in characteristic $p>0$, the rigidity of $\\overline{\\mathcal{M}}_{g,n}$, with the same exceptions as over $\\mathbb{C}$, implies that its automorphism group is $S_n$. We prove that, over any perfect field, $\\overline{M}_{0,n}$ is rigid and deduce that, over any field, $Aut(\\overline{M}_{0,n})\\cong S_{n}$ for $n\\geq 5$. Going back to characteristic zero, we prove that for $g+n>4$, the coarse moduli space $\\overline M_{g,n}$ is rigid, extending a result of Hacking who had proven it has no locally trivial deformations. Finally, we show that $\\overline{M}_{1,2}$ is not rigid, although it does not admit locally trivial deformations, by explicitly computing his Kuranishi family.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-07-08T22:15:37.000Z",
      "abstract": "The stack $\\overline{\\mathcal{M}}_{g,n}$ parametrizing Deligne-Mumford stable curves and its coarse moduli space $\\overline{M}_{g,n}$ are defined over $\\mathbb{Z}$, and therefore over any field. We investigate the rigidity of $\\overline{\\mathcal{M}}_{g,n}$, both in the sense of the absence of first order infinitesimal deformations and of automorphisms not coming form the permutations of the marked points. In particular, we prove that over any perfect field, $\\overline{M}_{0,n}$ does not have non-trivial first order infinitesimal deformations and we apply this result to show that, over any field, $Aut(\\overline{M}_{0,n})\\cong S_{n}$ for $n\\geq 5$. Furthermore, we extend some of these results to the stacks $\\overline{\\mathcal{M}}_{g,A[n]}$ parametrizing weighted stable curves and to their coarse moduli spaces $\\overline{M}_{g,A[n]}$. These spaces have been introduced by Hassett as compactifications of $\\mathcal{M}_{g,n}$ and $M_{g,n}$ respectively, by assigning rational weights $A = (a_{1},...,a_{n})$, $0< a_{i} \\leq 1$ to the markings. In particular we prove that $\\overline{M}_{0,A[n]}$ is rigid over any perfect field and that $\\overline{\\mathcal{M}}_{g,A[n]}$ for $g\\geq 1$ is rigid over any field of characteristic zero. Finally, we study the infinitesimal deformations of the coarse moduli space $\\overline{M}_{g,A[n]}$. We prove that over any field of characteristic zero $\\overline{M}_{g,A[n]}$ does not have locally trivial first order infinitesimal deformations if $g+n\\geq 4$. Furthermore we show that $\\overline{M}_{1,2}$ does not admit locally trivial deformations and that it is not rigid.",
      "comment": "30 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-11-08T18:18:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14H10",
      "14D22",
      "14D23",
      "14D06"
    ],
    "keywords": [
      "coarse moduli space",
      "arbitrary characteristic",
      "non-trivial first order infinitesimal deformations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1407.2284F"
    }
  }
}