{
  "id": "2001.02303",
  "version": "v1",
  "published": "2020-01-07T22:23:28.000Z",
  "updated": "2020-01-07T22:23:28.000Z",
  "title": "Néron models of intermediate Jacobians associated to moduli spaces",
  "authors": [
    "Ananyo Dan",
    "Inder Kaur"
  ],
  "comment": "to appear in Revista Matem\\'atica Complutense",
  "doi": "10.1007/s13163-019-00333-y",
  "categories": [
    "math.AG"
  ],
  "abstract": "Let $\\pi_1:\\mathcal{X} \\to \\Delta$ be a flat family of smooth, projective curves of genus $g \\ge 2$, degenerating to an irreducible nodal curve $X_0$ with exactly one node. Fix an invertible sheaf $\\mathcal{L}$ on $\\mathcal{X}$ of relative odd degree. Let $\\pi_2:\\mathcal{G}(2,\\mathcal{L}) \\to \\Delta$ be the relative Gieseker moduli space of rank $2$ semi-stable vector bundles with determinant $\\mathcal{L}$ over $\\mathcal{X}$. Since $\\pi_2$ is smooth over $\\Delta^*$, there exists a canonical family $\\widetilde{\\rho}_i:\\mathbf{J}^i_{\\mathcal{G}(2, \\mathcal{L})_{\\Delta^*}} \\to \\Delta^{*}$ of $i$-th intermediate Jacobians i.e., for all $t \\in \\Delta^*$, $(\\widetilde{\\rho}_i)^{-1}(t)$ is the $i$-th intermediate Jacobian of $\\pi_2^{-1}(t)$. There exist different N\\'{e}ron models $\\overline{\\rho}_i:\\overline{\\mathbf{J}}_{\\mathcal{G}(2, \\mathcal{L})}^i \\to \\Delta$ extending $\\widetilde{\\rho}_i$ to the entire disc $\\Delta$, constructed by Clemens, Saito, Schnell, Zucker and Green-Griffiths-Kerr. In this article, we prove that in our setup, the N\\'{e}ron model $\\overline{\\rho}_i$ is canonical in the sense that the different N\\'{e}ron models coincide and is an analytic fiber space which graphs admissible normal functions. We also show that for $1 \\le i \\le \\max\\{2,g-1\\}$, the central fiber of $\\overline{\\rho}_i$ is a fibration over product of copies of $J^k(\\mathrm{Jac}(\\widetilde{X}_0))$ for certain values of $k$, where $\\widetilde{X}_0$ is the normalization of $X_0$. In particular, for $g \\ge 5$ and $i=2, 3, 4$, the central fiber of $\\overline{\\rho}_i$ is a semi-abelian variety. Furthermore, we prove that the $i$-th generalized intermediate Jacobian of the (singular) central fibre of $\\pi_2$ is a fibration over the central fibre of the N\\'{e}ron model $\\overline{\\mathbf{J}}^i_{\\mathcal{G}(2, \\mathcal{L})}$. In fact, for $i=2$ the fibration is an isomorphism.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-01-07T22:23:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14C30",
      "14C34",
      "14D07",
      "32G20",
      "32S35",
      "14D20",
      "14H40"
    ],
    "keywords": [
      "néron models",
      "th intermediate jacobian",
      "central fibre",
      "central fiber",
      "th generalized intermediate jacobian"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Springer"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}