{
  "id": "1901.08111",
  "version": "v1",
  "published": "2019-01-23T20:00:39.000Z",
  "updated": "2019-01-23T20:00:39.000Z",
  "title": "An invariant detecting rational singularities via the log canonical threshold",
  "authors": [
    "Raf Cluckers",
    "Mircea Mustata"
  ],
  "comment": "16 pages",
  "categories": [
    "math.AG"
  ],
  "abstract": "We show that if f is a nonzero, noninvertible function on a smooth complex variety X and J_f is the Jacobian ideal of f, then lct(f, J_f^2)>1 if and only if the hypersurface defined by f has rational singularities. Moreover, if this is not the case, then lct(f, J_f^2)=lct(f). We give two proofs, one relying on arc spaces and one that shows that the minimal exponent of f is at least as large as lct(f, J_f^2). In the case of a polynomial over the algebraic closure of Q, we also prove an analogue of this latter inequality, with the minimal exponent replaced by the motivic oscillation index moi(f).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-01-23T20:00:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14B05",
      "14E18",
      "14J17",
      "11L07"
    ],
    "keywords": [
      "invariant detecting rational singularities",
      "log canonical threshold",
      "motivic oscillation index moi",
      "minimal exponent",
      "smooth complex variety"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}