{
  "id": "1506.07003",
  "version": "v1",
  "published": "2015-06-23T13:56:55.000Z",
  "updated": "2015-06-23T13:56:55.000Z",
  "title": "Graphs of Schemes Associated to Group Actions",
  "authors": [
    "Ali Ulas Ozgur Kisisel",
    "Engin Ozkan"
  ],
  "comment": "11 pages",
  "categories": [
    "math.AG",
    "math.AC"
  ],
  "abstract": "Let $X$ be a proper algebraic scheme over an algebraically closed field. We assume that a torus $T$ acts on $X$ such that the action has isolated fixed points. The $T$-graph of $X$ can be defined using the fixed points and the one dimensional orbits of the $T$-action. If the upper Borel subgroup of the general linear group with maximal torus $T$ acts on $X$, then we can define a second graph associated to $X$, called the $A$-graph of $X$. We prove that the $A$-graph of $X$ is connected if and only if $X$ is connected. We use this result to give a proof of Hartshorne's theorem on the connectedness of Hilbert scheme in the case of $d$ points in $\\mathbb{P}^{n}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-06-23T13:56:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14L30"
    ],
    "keywords": [
      "group actions",
      "proper algebraic scheme",
      "general linear group",
      "fixed points",
      "upper borel subgroup"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}