{
  "id": "0907.0302",
  "version": "v5",
  "published": "2009-07-02T07:31:18.000Z",
  "updated": "2011-01-22T19:59:15.000Z",
  "title": "Gröbner strata in the Hilbert scheme of points",
  "authors": [
    "Mathias Lederer"
  ],
  "comment": "47 pages, 6 figures, 11 examples",
  "categories": [
    "math.AG",
    "math.AC"
  ],
  "abstract": "The present paper shall provide a framework for working with Gr\\\"obner bases over arbitrary rings $k$ with a prescribed finite standard set $\\Delta$. We show that the functor associating to a $k$-algebra $B$ the set of all reduced Gr\\\"obner bases with standard set $\\Delta$ is representable and that the representing scheme is a locally closed stratum in the Hilbert scheme of points. We cover the Hilbert scheme of points by open affine subschemes which represent the functor associating to a $k$-algebra $B$ the set of all border bases with standard set $\\Delta$ and give reasonably small sets of equations defining these schemes. We show that the schemes parametrizing Gr\\\"obner bases are connected; give a connectedness criterion for the schemes parametrizing border bases; and prove that the decomposition of the Hilbert scheme of points into the locally closed strata parametrizing Gr\\\"obner bases is not a stratification.",
  "revisions": [
    {
      "version": "v5",
      "updated": "2011-01-22T19:59:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14C05",
      "13F20",
      "13P10"
    ],
    "keywords": [
      "hilbert scheme",
      "gröbner strata",
      "locally closed stratum",
      "open affine subschemes",
      "prescribed finite standard set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 47,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0907.0302L"
    }
  }
}