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.