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.