{
  "id": "1310.5387",
  "version": "v4",
  "published": "2013-10-21T00:19:15.000Z",
  "updated": "2015-06-30T00:24:28.000Z",
  "title": "On general fibers of Gauss maps in positive characteristic",
  "authors": [
    "Katsuhisa Furukawa"
  ],
  "comment": "8 pages",
  "categories": [
    "math.AG"
  ],
  "abstract": "A general fiber of the Gauss map of a projective variety in $\\mathbb{P}^N$ coincides with a linear subvariety of $\\mathbb{P}^N$ in characteristic zero. In positive characteristic, S. Fukasawa showed that a general fiber of the Gauss map can be a non-linear variety. In this paper, we show that each irreducible component of such a possibly non-linear fiber of the Gauss map is contracted to one point by the degeneracy map, and is contained in a linear subvariety corresponding to the kernel of the differential of the Gauss map. We also show the inseparability of Gauss maps of strange varieties not being cones.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2014-07-24T07:19:00.000Z",
      "title": "On general fibers of Gauss maps in arbitrary characteristic",
      "abstract": "We investigate general fibers of the Gauss map $\\gamma$ of a projective variety $X$ in $\\mathbb{P}^N$, in terms of the degeneracy map $\\kappa$ from $X$ to a Grassmann variety. Here, if $\\gamma$ is separable, then a general fiber of $\\gamma$ coincides with the linear subvariety $\\kappa(x)$ of $\\mathbb{P}^N$ for general $x$ in the fiber. In this paper, we show the following results: (1) Equivalence of the separability of the two maps $\\gamma$ and $\\kappa$. (2) $\\kappa(x)$ is constant on $x \\in F$ for each irreducible component $F$ of a general fiber of $\\gamma$; in particular, $F$ is contained in this constant linear subvariety even if $\\gamma$ is inseparable. (3) Inseparability of Gauss maps of strange varieties; this is useful to construct some typical examples.",
      "comment": "21 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v4",
      "updated": "2015-06-30T00:24:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14N05",
      "14M15"
    ],
    "keywords": [
      "general fiber",
      "gauss map",
      "arbitrary characteristic",
      "constant linear subvariety",
      "grassmann variety"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1310.5387F"
    }
  }
}