{
  "id": "1606.09625",
  "version": "v1",
  "published": "2016-06-30T19:28:34.000Z",
  "updated": "2016-06-30T19:28:34.000Z",
  "title": "Conjugacy classes of commuting nilpotents",
  "authors": [
    "William Haboush",
    "Donghoon Hyeon"
  ],
  "comment": "17 pages",
  "categories": [
    "math.AG"
  ],
  "abstract": "We consider the space $\\mathcal M_{q,n}$ of regular $q$-tuples of commuting nilpotent endomorphisms of $k^n$ modulo simultaneous conjugation. We show that $\\mathcal M_{q,n}$ admits a natural homogeneous space structure, and that it is an affine space bundle over $\\mathbb P^{q-1}$. A closer look at the homogeneous structure reveals that, over $\\mathbb C$ and with respect to the complex topology, $\\mathcal M_{q,n}$ is a smooth vector bundle over $\\mathbb P^{q-1}$. We prove that, in this case, $\\mathcal M_{q,n}$ is diffeomorphic to a direct sum of twisted tangent bundles. We also prove that $\\mathcal M_{q,n}$ possesses a universal property and represents a functor of ideals, and use it to identify $\\mathcal M_{q,n}$ with an open subscheme of a punctual Hilbert scheme. By using a result of A. Iarrobino, we show that $\\mathcal M_{q,n} \\to \\mathbb P^{q-1}$ is not a vector bundle, hence giving a family of affine space bundles that are not vector bundles.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-06-30T19:28:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14D06",
      "14D22",
      "14R20",
      "14C05"
    ],
    "keywords": [
      "conjugacy classes",
      "affine space bundle",
      "natural homogeneous space structure",
      "smooth vector bundle",
      "punctual hilbert scheme"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}