{
  "id": "math/0305286",
  "version": "v2",
  "published": "2003-05-20T12:44:37.000Z",
  "updated": "2003-12-03T02:37:18.000Z",
  "title": "F-singularities of pairs and Inversion of Adjunction of arbitrary codimension",
  "authors": [
    "Shunsuke Takagi"
  ],
  "comment": "21 pages, AMS-LaTeX; v.2: minor changes, to appear in Invent. Math",
  "doi": "10.1007/s00222-003-0350-3",
  "categories": [
    "math.AG",
    "math.AC"
  ],
  "abstract": "We generalize the notions of F-regular and F-pure rings to pairs $(R,\\a^t)$ of rings $R$ and ideals $\\a \\subset R$ with real exponent $t > 0$, and investigate these properties. These ``F-singularities of pairs'' correspond to singularities of pairs of arbitrary codimension in birational geometry. Via this correspondence, we prove Inversion of Adjunction of arbitrary codimension, which states that for a pair $(X,Y)$ of a smooth variety $X$ and a closed subscheme $Y \\subsetneq X$, if the restriction $(Z, Y|_Z)$ to a normal $\\Q$-Gorenstein closed subvariety $Z \\subsetneq X$ is klt (resp. lc), then the pair $(X,Y+Z)$ is plt (resp. lc) near $Z$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2003-12-03T02:37:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "13A35",
      "14B05",
      "14E15",
      "14E30"
    ],
    "keywords": [
      "arbitrary codimension",
      "f-singularities",
      "adjunction",
      "gorenstein closed subvariety",
      "f-pure rings"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "journal": "Inventiones Mathematicae",
      "year": 2004,
      "month": "Jul",
      "volume": 157,
      "number": 1,
      "pages": 123
    },
    "note": {
      "typesetting": "LaTeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004InMat.157..123T"
    }
  }
}