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Néron models of intermediate Jacobians associated to moduli spaces

Let $π_1:\mathcal{X} \to Δ$ 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 $π_2:\mathcal{G}(2,\mathcal{L}) \to Δ$ be the relative Gieseker moduli space of rank $2$ semi-stable vector bundles with determinant $\mathcal{L}$ over $\mathcal{X}$. Since $π_2$ is smooth over $Δ^*$, there exists a canonical family $\widetildeρ_i:\mathbf{J}^i_{\mathcal{G}(2, \mathcal{L})_{Δ^*}} \to Δ^{*}$ of $i$-th intermediate Jacobians i.e., for all $t \in Δ^*$, $(\widetildeρ_i)^{-1}(t)$ is the $i$-th intermediate Jacobian of $π_2^{-1}(t)$. There exist different Néron models $\overlineρ_i:\overline{\mathbf{J}}_{\mathcal{G}(2, \mathcal{L})}^i \to Δ$ extending $\widetildeρ_i$ to the entire disc $Δ$, constructed by Clemens, Saito, Schnell, Zucker and Green-Griffiths-Kerr. In this article, we prove that in our setup, the Néron model $\overlineρ_i$ 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 $\overlineρ_i$ 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 $\overlineρ_i$ is a semi-abelian variety. Furthermore, we prove that the $i$-th generalized intermediate Jacobian of the (singular) central fibre of $π_2$ 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.