{
  "id": "1208.3317",
  "version": "v2",
  "published": "2012-08-16T08:59:01.000Z",
  "updated": "2015-03-29T20:17:32.000Z",
  "title": "Rational curves on \\bar{M}_g and K3 surfaces",
  "authors": [
    "Luca Benzo"
  ],
  "comment": "published version, revisions in the exposition, minor mistakes corrected",
  "journal": "International Mathematics Research Notices, Volume 15 (2014), pp. 4179-4214",
  "doi": "10.1093/imrn/rnt067",
  "categories": [
    "math.AG"
  ],
  "abstract": "Let $(S,L)$ be a smooth primitively polarized K3 surface of genus $g$ and $f:X \\rightarrow \\mathbb{P}^1$ the fibration defined by a linear pencil in $|L|$. For $f$ general and $g \\geq 7$, we work out the splitting type of the locally free sheaf $\\Psi^{*}_f T_{{\\overline{M}}_g}$, where $\\Psi_f$ is the modular morphism associated to $f$. We show that this splitting type encodes the fundamental geometrical information attached to Mukai's projection map $\\mathcal{P}_g \\rightarrow \\overline{\\mathcal{M}}_g$, where $\\mathcal{P}_g$ is the stack parameterizing pairs $(S,C)$ with $(S,L)$ as above and $C \\in |L|$ a stable curve. Moreover, we work out conditions on a fibration $f$ to induce a modular morphism $\\Psi_f$ such that the normal sheaf $N_{\\Psi_f}$ is locally free.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-08-16T08:59:01.000Z",
      "abstract": "We work out the splitting type of the locally free sheaf $\\Psi^{*}_f T_{{\\bar{M}}_g}$, $g \\geq 7$, where $\\Psi_f$ is the modular morphism associated to a fibration $f:X \\rightarrow \\mathbb{P}^1$ whose general fibre is a general curve in $\\mathcal{K}_g$, the locus of smooth curves of genus $g$ lying on a K3 surface. We show that this splitting type encodes the fundamental geometrical informations attached to Mukai's projection map $\\mathcal{P}_g \\rightarrow \\mathcal{K}_g$. Moreover we work out conditions on a fibration $f$ to induce a modular morphism $\\Psi_f$ such that the normal sheaf $N_{\\Psi_f}$ is locally free.",
      "comment": "35 pages",
      "journal": null
    },
    {
      "version": "v2",
      "updated": "2015-03-29T20:17:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14J28",
      "14H10",
      "14D06",
      "14D15"
    ],
    "keywords": [
      "k3 surface",
      "rational curves",
      "modular morphism",
      "splitting type",
      "mukais projection map"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1208.3317B"
    }
  }
}