{
  "id": "1003.4241",
  "version": "v6",
  "published": "2010-03-22T18:37:39.000Z",
  "updated": "2018-07-12T19:01:29.000Z",
  "title": "The Zariski-Lipman conjecture for complete intersections",
  "authors": [
    "Rolf Källström"
  ],
  "comment": "17 pages. A mistake in the previous version was discovered. The new version is a rather thorough rewrite",
  "categories": [
    "math.AG",
    "math.AC"
  ],
  "abstract": "The tangential ramification locus $B_{X/Y}^t\\subset B_{X/Y}$ is the subset of points in the ramification locus where the sheaf of relative vector fields $T_{X/Y}$ fails to be locally free. It was conjectured by Zariski and Lipman that if $V/k$ is a variety over a field $k$ of characteristic 0 and $B^t_{V/k}= \\emptyset$, then $V/k$ is smooth (=regular). We prove this conjecture when $V/k$ is a locally complete intersection. We prove also that $B_{V/k}^t= \\emptyset$ implies $\\operatorname {codim}_V B_{V/k}\\leq 1 $ in positive characteristic, if $V/k $ is the fibre of a flat morphism satisfying generic smoothness.",
  "revisions": [
    {
      "version": "v5",
      "updated": "2011-01-25T21:07:33.000Z",
      "abstract": "The tangential branch locus $B_{X/Y}^t\\subset B_{X/Y}$ is the subset of points in the branch locus where the sheaf of relative vector fields $T_{X/Y}$ fails to be locally free. It was conjectured by Zariski and Lipman that if $V/k$ is a variety over a field $k$ of characteristic 0 and $B^t_{V/k}= \\emptyset$, then $V/k$ is smooth (=regular). We prove this conjecture when $V/k$ is a locally complete intersection. We prove also that $B_{V/k}^t= \\emptyset$ implies $\\codim_X B_{V/k}\\leq 1 $ in positive characteristic, if $V/k $ is the fibre of a flat morphism satisfying generic smoothness.",
      "comment": "15 pages, the use of flatness is clarified. Improved presentation",
      "journal": null,
      "doi": null
    },
    {
      "version": "v6",
      "updated": "2018-07-12T19:01:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14A10",
      "32C38"
    ],
    "keywords": [
      "complete intersection",
      "zariski-lipman conjecture",
      "flat morphism satisfying generic smoothness",
      "tangential branch locus",
      "vector fields"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1003.4241K"
    }
  }
}