{
  "id": "1903.06377",
  "version": "v1",
  "published": "2019-03-15T06:10:05.000Z",
  "updated": "2019-03-15T06:10:05.000Z",
  "title": "The Hilbert scheme of a pair of linear spaces",
  "authors": [
    "Ritvik Ramkumar"
  ],
  "comment": "32 pages, 1 figure; comments are always welcome!",
  "categories": [
    "math.AG",
    "math.AC"
  ],
  "abstract": "Let $H(c,d,n)$ be the component of the Hilbert scheme whose general point parameterizes a $c$-plane union a $d$-plane in $\\mathbf{P}^n$. We show that $H(c,d,n)$ is non-singular and isomorphic to successive blow ups of $\\mathbf{G}(c,n) \\times \\mathbf{G}(d,n)$ or $\\text{Sym}^{2}\\mathbf{G}(d,n)$. We also show that $H(c,d,n)$ has exactly one borel fixed point, describe the finitely many orbits under the $\\text{PGL}_n$ action and describe the effective and ample cones. Using this, we prove that $H(n-3,n-3,n)$ and $H(1,n-2,n)$ are Mori dream spaces. In the case of $H(1,n-2,n)$, we show that the full Hilbert scheme contains only one more component and we give a complete description of its singularities. We end by describing the singularities of some Hilbert schemes parameterizing more than two linear spaces.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-03-15T06:10:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "13D02",
      "13D10",
      "14D22",
      "14E05",
      "14E30",
      "14M15"
    ],
    "keywords": [
      "linear spaces",
      "full hilbert scheme contains",
      "mori dream spaces",
      "general point parameterizes",
      "complete description"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}