{
  "id": "1305.6268",
  "version": "v1",
  "published": "2013-05-27T16:13:11.000Z",
  "updated": "2013-05-27T16:13:11.000Z",
  "title": "A geometric definition of Gabrielov numbers",
  "authors": [
    "Wolfgang Ebeling",
    "Atsushi Takahashi"
  ],
  "comment": "13 pages, 6 figures",
  "categories": [
    "math.AG"
  ],
  "abstract": "Gabrielov numbers describe certain Coxeter-Dynkin diagrams of the 14 exceptional unimodal singularities and play a role in Arnold's strange duality. In a previous paper, the authors defined Gabrielov numbers of a cusp singularity with an action of a finite abelian subgroup $G$ of ${\\rm SL}(3,\\CC)$ using the Gabrielov numbers of the cusp singularity and data of the group $G$. Here we consider a crepant resolution $Y \\to \\CC^3/G$ and the preimage $Z$ of the image of the Milnor fibre of the cusp singularity under the natural projection $\\CC^3 \\to \\CC^3/G$. Using the McKay correspondence, we compute the homology of the pair $(Y,Z)$. We construct a basis of the relative homology group $H_3(Y,Z;\\QQ)$ with a Coxeter-Dynkin diagram where one can read off the Gabrielov numbers.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-05-27T16:13:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "32S25",
      "32S55",
      "14E16",
      "14L30"
    ],
    "keywords": [
      "geometric definition",
      "cusp singularity",
      "coxeter-dynkin diagram",
      "authors defined gabrielov numbers",
      "exceptional unimodal singularities"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1305.6268E"
    }
  }
}