Néron models of intermediate Jacobians associated to moduli spaces

Abstract

Let ${\pi }_1:\mathcal{X}\to \Delta$ 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 ${\pi }_2:\mathcal{G}(2,\mathcal{L})\to \Delta$ 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 ${\Delta }^{\ast }$, there exists a canonical family ${\tilde{\rho }}_i:{\mathbf{J}}_{\mathcal{G}(2,\mathcal{L}{)}_{{\Delta }^{\ast }}}^i\to {\Delta }^{\ast }$ of $i$-th intermediate Jacobians i.e., for all $t\in {\Delta }^{\ast }$, $({\tilde{\rho }}_i{)}^{-1}(t)$ is the $i$-th intermediate Jacobian of ${\pi }_2^{-1}(t)$. There exist different Néron models ${\overline{\rho }}_i:{\overline{\mathbf{J}}}_{\mathcal{G}(2,\mathcal{L})}^i\to \Delta$ extending ${\tilde{\rho }}_i$ 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 $J^k(\mathrm{Jac}({\tilde{X}}_0))$ for certain values of $k$, where ${\tilde{X}}_0$ 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 ${\overline{\mathbf{J}}}_{\mathcal{G}(2,\mathcal{L})}^i$. In fact, for $i=2$ the fibration is an isomorphism.

to appear in Revista Matemática Complutense

References in corpus (1)

• Existence of semistable vector bundles with fixed determinants