Néron models of intermediate Jacobians associated to moduli spaces

Abstract

Let \(\pi _1:\mathcal {X} \rightarrow \Delta \) 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 \(\pi _2:\mathcal {G}(2,\mathcal {L}) \rightarrow \Delta \) 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 \(\widetilde{\rho }_i:\mathbf {J}^i_{\mathcal {G}(2, \mathcal {L})_{\Delta ^*}} \rightarrow \Delta ^{*}\) 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 Néron models \(\overline{\rho }_i:\overline{\mathbf {J}}_{\mathcal {G}(2, \mathcal {L})}^i \rightarrow \Delta \) 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 Néron model \(\overline{\rho }_i\) is canonical in the sense that the different Néron 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 Néron model \(\overline{\mathbf {J}}^i_{\mathcal {G}(2, \mathcal {L})}\). In fact, for \(i=2\) the fibration is an isomorphism.

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Abbreviations

\(X_0,x_0\) :

Irreducible nodal curve \(X_0\) with node at \(x_0\)

\(\pi : \widetilde{X}_0 \rightarrow X_0\) :

Normalization of \(X_0\)

\(\Delta , \Delta ^*\) :

Open, unit disc \(\Delta \) and \(\Delta ^*:=\Delta \backslash \{0\}\)

\(\rho : \mathcal {Y} \rightarrow \Delta \) :

Family of projective varieties, smooth over \(\Delta ^*\)

\(\mathcal {Y}_t\) :

The fiber \(\rho ^{-1}(t)\) for any \(t \in \Delta \)

\(\mathcal {Y}_\infty \) :

The base change of the family \(\rho \) under the natural morphism \(\mathfrak {h} \rightarrow \Delta ^* \hookrightarrow \Delta \), where \(\mathfrak {h}\) is the universal covering of \(\Delta ^*\)

\(\mathcal {Y}_{\Delta ^*}\) :

restriction of \(\mathcal {Y}\) to \(\Delta ^*\)

\(\mathcal {H}^i_{\mathcal {Y}_{\Delta ^*}}, F^p\mathcal {H}^i_{\mathcal {Y}_{\Delta ^*}}\) :

Hodge bundles associated to the family \(\mathcal {Y}_{\Delta ^*}\)

\(\overline{\mathcal {H}}^i_{\mathcal {Y}_{\Delta ^*}}, F^p\overline{\mathcal {H}}^i_{\mathcal {Y}_{\Delta ^*}}\) :

Canonical extensions of \(\mathcal {H}^i_{\mathcal {Y}_{\Delta ^*}}, F^p\mathcal {H}^i_{\mathcal {Y}_{\Delta ^*}}\), respectively

\(\widetilde{\rho }: \mathbf {J}^i_{\mathcal {Y}_{\Delta ^*}} \rightarrow \Delta ^*\) :

Family of i-th intermediate Jacobians associated to \(\mathcal {Y}_{\Delta ^*}\)

\(\overline{\rho }: \overline{\mathbf {J}}^i_{\mathcal {Y}} \rightarrow \Delta \) :

Néron model associated to \(\widetilde{\rho }\)

\(T_{s,i}, T_{s,i}^{\mathbb {Q}}\) :

Local monodromy transformation associated to \(\rho \)

\(T_i:H^i(\mathcal {Y}_\infty , \mathbb {Q}) \rightarrow H^i(\mathcal {Y}_\infty , \mathbb {Q})\) :

Limit monodromy transformation

\(N_i\) :

\(\log (T_i)\)

\(\mathrm {sp}_i: H^i(\mathcal {Y}_0,\mathbb {Z}) \rightarrow H^i(\mathcal {Y}_\infty , \mathbb {Z})\) :

Specialization morphism

\(M_Y(2,\mathcal {L}')\) :

Moduli space of rank 2, semi-stable sheaves with determinant \(\mathcal {L}'\) over Y

\(\pi _1: \mathcal {X} \rightarrow \Delta \) :

Family of projective curves with central fiber \(X_0\), smooth over \(\Delta ^*\)

\(\mathcal {L}, \mathcal {L}_0, \widetilde{\mathcal {L}}_0\) :

Odd degree invertible sheaf \(\mathcal {L}\) on \(\mathcal {X}\), \(\mathcal {L}_0:=\mathcal {L}|_{X_0}\), \(\widetilde{\mathcal {L}}_0:=\pi ^*\mathcal {L}_0\)

\(\widetilde{\pi }_1: \widetilde{\mathcal {X}} \rightarrow \mathcal {X} \xrightarrow {\pi } \Delta \) :

Blow-up of \(\mathcal {X}\) at \(x_0\)

\(\pi _2: \mathcal {G}(2,\mathcal {L}) \rightarrow \Delta \) :

Relative Gieseker moduli space associated to \(\pi _1\)

\(\mathcal {G}_{X_0}(2,\mathcal {L}_0)\) :

Central fiber of the moduli space \(\mathcal {G}(2,\mathcal {L})\)

\(\mathcal {G}_0, \mathcal {G}_1\) :

The two irreducible components of \(\mathcal {G}_{X_0}(2,\mathcal {L}_0)\)

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Acknowledgements

We thank Prof. J. F. de Bobadilla, Dr. B. Sigurdsson and Dr. S. Basu for numerous discussions. The first author is currently supported by ERCEA Consolidator Grant 615655-NMST and also by the Basque Government through the BERC \(2014-2017\) program and by Spanish Ministry of Economy and Competitiveness MINECO: BCAM Severo Ochoa excellence accreditation SEV-\(2013-0323\). The second author is funded by CAPES-PNPD scholarship.

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Dan, A., Kaur, I. Néron models of intermediate Jacobians associated to moduli spaces. Rev Mat Complut 33, 885–910 (2020). https://doi.org/10.1007/s13163-019-00333-y

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Keywords

  • Torelli theorem
  • Intermediate Jacobians
  • Néron models
  • Nodal curves
  • Gieseker moduli space
  • Limit mixed Hodge structures

Mathematics Subject Classification

  • Primary 14C30
  • 14C34
  • 14D07
  • 32G20
  • 32S35
  • 14D20
  • Secondary 14H40