{
  "id": "math/0305221",
  "version": "v1",
  "published": "2003-05-15T18:43:41.000Z",
  "updated": "2003-05-15T18:43:41.000Z",
  "title": "Nonemptiness of skew-symmetric degeneracy loci",
  "authors": [
    "William Graham"
  ],
  "comment": "13 pages",
  "categories": [
    "math.AG"
  ],
  "abstract": "Let V be a rank N vector bundle on a d-dimensional complex projective scheme X; assume that V is equipped with a skew-symmetric bilinear form with values in a line bundle L and that \\Lambda^2 V^* \\otimes L is ample. Suppose that the maximum rank of the form at any point of X is r, where r>0 is even. The main result of this paper is that if d>2(N-r), then the locus of points where the rank of the form is at most r-2 is nonempty. This is a skew-symmetric analogue of the main result of math.AG/0305159; the proof is similar. If the hypothesis of ampleness is relaxed, we obtain a weaker estimate on the maximum dimension of X (and give a similar result for the symmetric case). We give applications to subschemes of skew-symmetric matrices, and to the stratification of the dual of a Lie algebra by orbit dimension.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2003-05-15T18:43:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14N05"
    ],
    "keywords": [
      "skew-symmetric degeneracy loci",
      "nonemptiness",
      "main result",
      "skew-symmetric bilinear form",
      "d-dimensional complex projective scheme"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2003math......5221G"
    }
  }
}