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Néron models of intermediate Jacobians associated to moduli spaces

Research paper by Ananyo Dan, Inder Kaur

Indexed on: 07 Nov '19Published on: 07 Nov '19Published in: Revista Matemática Complutense



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.