{
  "id": "1403.1304",
  "version": "v2",
  "published": "2014-03-06T00:51:49.000Z",
  "updated": "2015-04-07T23:36:07.000Z",
  "title": "On Chow Stability for algebraic curves",
  "authors": [
    "L. Brambila-Paz",
    "H. Torres-Lopez"
  ],
  "categories": [
    "math.AG"
  ],
  "abstract": "In the last decades there have been introduced different concepts of stability for projective varieties. In this paper we give a natural and intrinsic criterion of the Chow, and Hilbert, stability for complex irreducible smooth projective curves $C\\subset \\mathbb P ^n$. Namely, if the restriction $T\\mathbb P_{|C} ^n$ of the tangent bundle of $\\mathbb P ^n$ to $C$ is stable then $C\\subset \\mathbb P ^n$ is Chow stable, and hence Hilbert stable. We apply this criterion to describe a smooth open set of the irreducible component $Hilb^{P(t),s}_{{Ch}}$ of the Hilbert scheme of $\\mathbb{P} ^n$ containing the generic smooth Chow-stable curve of genus $g$ and degree $d>g+n-\\left\\lfloor\\frac{g}{n+1}\\right\\rfloor.$ Moreover, we describe the quotient stack of such curves. Similar results are obtained for the locus of Hilbert stable curves.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-03-06T00:51:49.000Z",
      "abstract": "In the last decades there have been introduced different concepts of stability for projective varieties. In this paper we give a natural and intrinsic criterion of the Chow, and Hilbert, stability for irreducible complete reduced curves $C$, with at most ordinary nodes and cusps as singularities, in a projective space $\\mathbb P ^n$. Namely, if the restriction $T\\mathbb P_{|C} ^n$ of the tangent bundle of $\\mathbb P ^n$ to $C$ is stable then $C\\subset \\mathbb P ^n$ is Chow stable, and hence Hilbert stable. We apply this criterion to describe a smooth open set of the irreducible component $Hilb^{s}_{{Ch}}$ of the Hilbert scheme of $\\mathbb{P} ^n$ containing the generic smooth Chow-stable curve of genus $g$ and degree $d>g+n-\\left\\lfloor\\frac{g}{n+1}\\right\\rfloor.$ Moreover, we describe the quotient stack of such curves. Similar results are obtained for the locus of Hilbert stable curves.",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-04-07T23:36:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14H60",
      "14H10",
      "14H40",
      "14H20",
      "14C05",
      "14D23"
    ],
    "keywords": [
      "algebraic curves",
      "chow stability",
      "smooth open set",
      "generic smooth chow-stable curve",
      "ordinary nodes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1403.1304B"
    }
  }
}