{
  "id": "2007.12119",
  "version": "v1",
  "published": "2020-07-23T16:45:18.000Z",
  "updated": "2020-07-23T16:45:18.000Z",
  "title": "Deformation and Unobstructedness of Determinantal Schemes",
  "authors": [
    "Jan O. Kleppe",
    "Rosa M. Miró-Roig"
  ],
  "comment": "Accepted for publication by the Memoirs of the American Mathematical Society",
  "categories": [
    "math.AG",
    "math.AC"
  ],
  "abstract": "Let $Hilb ^{p(t)}(P^n)$ be the Hilbert scheme of closed subschemes of $P^n$ with Hilbert polynomial $p(t) \\in Q[t]$, and let $W:= \\overline{W(\\underline{b};\\underline{a};r)}$ be the closure of the locus in $Hilb ^{p(t)}(P^n)$ of determinantal schemes defined by the vanishing of the $(t-r+1)\\times (t - r+1)$ minors of some matrix $\\mathcal A$ of size $t\\times (t+c-1)$ with $ij$-enty a homogeneous form of degree $a_j-b_i$ and with $r$ satisfying $\\max\\{1,2-c\\} \\le r < t$. $W$ is an irreducible algebraic set. First of all, we compute an upper $r$-independent bound for the dimension of $W$ in terms of $a_j$ and $b_i$ which is sharp for $r=1$. In the linear case ($a_j = 1, b_i=0$) and cases sufficiently close, we conjecture and to a certain degree prove that this bound is achieved for all $r$. Then, we study to what extent $W$ is a generically smooth component of $Hilb ^{p(t)}(P^n)$. Under some weak numerical assumptions on the integers $a_j$ and $b_i$ (or under some depth conditions) we conjecture and often prove that $W$ is a generically smooth component. Moreover, we also study the depth of the normal module of the homogeneous coordinate ring of $(X)\\in W$ and of a closely related module. We conjecture, and in some cases prove, that their codepth is often 1 (resp. $r$). These results extend previous results on standard determinantal schemes to determinantal schemes; i.e. previous results of the authors on $W(\\underline{b};\\underline{a};1)$ to $W$ with $1\\le r < t$ and $c\\ge 2-r$. Finally, deformations of exterior powers of the cokernel of the map determined by $\\mathcal A$ are studied and proven to be given as deformations of $X \\subset P^n$ if $\\dim X \\ge 3$. The work contains many examples which illustrate the results obtained and a considerable number of open problems; some of them are collected as conjectures in the final section.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-07-23T16:45:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14M12",
      "14C05",
      "14H10",
      "14J10",
      "13C40"
    ],
    "keywords": [
      "deformation",
      "generically smooth component",
      "unobstructedness",
      "conjecture",
      "standard determinantal schemes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}