{
  "id": "2006.14696",
  "version": "v1",
  "published": "2020-06-25T20:58:59.000Z",
  "updated": "2020-06-25T20:58:59.000Z",
  "title": "Complex surface singularities with rational homology disk smoothings",
  "authors": [
    "Jonathan Wahl"
  ],
  "comment": "27 pages, to be published in Proceedings of the N\\'{e}methi60 Conference",
  "categories": [
    "math.AG",
    "math.GT"
  ],
  "abstract": "A cyclic quotient singularity of type $p^2/pq-1$ ($0<q<p, (p,q)=1$) has a smoothing whose Milnor fibre is a $\\mathbb Q$HD, or rational homology disk (i.e., the Milnor number is $0$) ([9], 5.9.1). In the 1980's, we discovered additional examples of such singularities: three triply-infinite and six singly-infinite families, all weighted homogeneous. Later work of Stipsicz, Szab\\'{o}, Bhupal, and the author ([7], [1]) proved that these were the only weighted homogeneous examples. In his UNC PhD thesis (unpublished but available at [2]), our student Jacob Fowler completed the analytic classification of these singularities, and counted the number of smoothings in each case, except for types $\\mathcal W$, $\\mathcal N$, and $\\mathcal M$. In this paper, we describe his results, and settle these remaining cases; there is a unique $\\mathbb Q$HD smoothing component except in the cases of an obvious symmetry of the resolution dual graph. The method involves study of configurations of rational curves on projective rational surfaces.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-06-25T20:58:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14J17",
      "32S30",
      "14B07",
      "14J26"
    ],
    "keywords": [
      "rational homology disk smoothings",
      "complex surface singularities",
      "cyclic quotient singularity",
      "resolution dual graph",
      "student jacob fowler"
    ],
    "tags": [
      "conference paper"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}