{
  "id": "1210.3841",
  "version": "v2",
  "published": "2012-10-14T20:56:40.000Z",
  "updated": "2013-11-15T16:02:54.000Z",
  "title": "Hilbert series of certain jet schemes of determinantal varieties",
  "authors": [
    "Sudhir R. Ghorpade",
    "Boyan Jonov",
    "B. A. Sethuraman"
  ],
  "categories": [
    "math.CO",
    "math.AC",
    "math.AG"
  ],
  "abstract": "We consider the affine variety ${\\mathcal{Z}_{2,2}^{m,n}}$ (or just \"$Y$\") of first order jets over ${\\mathcal{Z}_{2}^{m,n}}$ (or just \"$X$\"), where $X$ is the classical determinantal variety given by the vanishing of all $2\\times 2$ minors of a generic $m\\times n$ matrix. When $2 < m \\le n$, this jet scheme $Y$ has two irreducible components: a trivial component, isomorphic to an affine space, and a nontrivial component that is the closure of the jets supported over the smooth locus of $X$. This second component is referred to as the principal component of $Y$; it is, in fact, a cone and can also be regarded as a projective subvariety of $\\mathbf{P}^{2mn-1}$. We prove that the degree of the principal component of $Y$ is the square of the degree of $X$ and more generally, the Hilbert series of the principal component of $Y$ is the square of the Hilbert series of $X$. As an application, we compute the $a$-invariant of the principal component of $Y$ and show that the principal component of $Y$ is Gorenstein if and only if $m=n$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-11-15T16:02:54.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "hilbert series",
      "principal component",
      "jet scheme",
      "first order jets",
      "affine variety"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1210.3841G"
    }
  }
}