论文标题
与模量空间相关的中间雅各布人的néron模型
Néron models of intermediate Jacobians associated to moduli spaces
论文作者
论文摘要
令$π_1：\ Mathcal {x} \toδ$是一个平稳的，一组平滑，投射曲线的家族，$ g \ ge 2 $，退化为不可减少的节点曲线$ x_0 $，恰好一个节点。将可逆的捆$ \ MATHCAL {l} $上的$ \ Mathcal {x} $的相对奇数度固定。令$π_2：\ MATHCAL {G}（2，\ Mathcal {l}）\toδ$是$ 2 $ 2 $半稳定矢量套件的相对Gieseker Moduli，带有$ \ Mathcal {l} $，超过$ \ nathcal {x} $。由于$π_2$在$δ^*$上平稳，因此存在一个规范的家庭$ \wideTildeρ_i：\ MathBf {J}^i _ {\ Mathcal {\ Mathcal {g}（2，\ Mathcal {l} Δ^*$，$（\widetildeρ_i）^{ - 1}（t）$的全部$ t \是$ i $ -th的$π_2^{ - 1}（t）$的$ i $ -th Intermediate Jacobian。存在不同的néron模型$ \overlineρ_i：\ overline {\ mathbf {j}} _ {\ Mathcal {\ mathcal {g}（2，2，\ mathcal {l}）}^i \toΔ$ toΔ$扩展$ \ \ \ \\ wideTildeC_i $ to to to to piss $Δ$，cly zuck y zuck a zuck y zuck和schn schn clemens and zut sait，sait sait sait sait sait，sait sait，sait s，绿色 - 长石 - 凯尔。在本文中，我们证明，在我们的设置中，néron模型$ \overlineρ_i$在不同的néron模型重合的意义上是规范的，并且是一个分析纤维空间，可以绘制可允许的正常函数。我们还表明，价格为$ 1 \ le i \ le \ max \ {2，g-1 \} $，$ \ overlineρ_i$的中央光纤是$ j^k（\ sathrm {jac}（\ widetilde {x} _0）$的$ j^k（\ mathrm {jac} $ k $ k $ k $ k $ k $ k $ k $ k $ k $ k $ k $ k $ k $ k $ k $ k $ k $） $ x_0 $的归一化。特别是，对于$ g \ ge 5 $和$ i = 2、3、4 $，$ \overlineρ_i$的中央光纤是半阿布莱的品种。此外，我们证明（$π_2$的（单数）中心纤维的$ i $ th的$ i $ themization jacobian是néron型号$ \ overline {\ mathbf {j}}^i_ _ {\ nathcal {\ mathcal {g}（g}（2，2，y Mathcal）的中心纤维的纤维。实际上，对于$ i = 2 $，振动是同构。
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.
