{
  "id": "math/0007156",
  "version": "v1",
  "published": "2000-07-26T12:34:42.000Z",
  "updated": "2000-07-26T12:34:42.000Z",
  "title": "Painlevé equations and deformations of rational surfaces with rational double points",
  "authors": [
    "Masa-Hiko Saito",
    "Hiroshi Umemura"
  ],
  "comment": "45 pages",
  "journal": "Physics and combinatorics 1999 (Nagoya), 320--365, World Sci. Publishing, River Edge, NJ, 2001.",
  "categories": [
    "math.AG"
  ],
  "abstract": "In this paper, we show that the B\\\"acklund transformations of Painlev\\'e equations come from birational maps of rational surfaces constructed by Okamoto as the spaces of initial conditions. The simultaneous resolutions of rational double points of the families of rational surfaces give rise to flops which are origins of symmetry groups of Painlev\\'e equation. Moreover we point out that Kodaira--Spencer theory of deformations of rational surfaces explains a geomrtric meaning of Painlev\\'e equations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2000-07-26T12:34:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14D15",
      "34M55",
      "32G10",
      "14J26"
    ],
    "keywords": [
      "rational double points",
      "deformations",
      "painleve equations come",
      "rational surfaces explains",
      "birational maps"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 45,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}