将Andreotti对Torelli定理的证明推广到节点曲线
Extending Andreotti's proof of Torelli's theorem to nodal curves
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中文总结 AI 辅助
本文推广Andreotti的证明到节点曲线，证明稳定半阿贝尔对同构可函子地诱导典范三元组间的射影等价，恢复紧化Torelli定理，并修复经典证明的技术缺陷。
中文摘要 AI 辅助
设$C$是特征为零的代数闭域$k$上亏格为$g$的连通节点曲线。令$A(C)=\Big(J(C), {\overline{P^{g-1}_C}}, \Theta(C) \Big)$为与其在$g-1$次的典范紧化雅可比簇$\overline{P^{g-1}_C}$相关联的稳定半阿贝尔对（在Alexeev的意义下）。$C$的典范三元组由其典范像的闭包以及两组区分点组成，这两组点编码了非分离节点的像和典范映射的分支数据的信息。我们证明了一个函子性重构：给定$k$上的连通节点曲线$C$和$C'$，稳定半阿贝尔对的每个同构$A(C)\cong A(C')$都典范地诱导出$C$和$C'$的相应典范三元组之间的一个射影等价。通过同构类，这恢复了Caporaso-Viviani的紧化Torelli定理中的重构陈述。我们的证明推广了Andreotti的高斯映射策略。从高斯图的规范化的投影的分支轨迹通过射影对偶性恢复了非线性典范分量和分支数据；紧化雅可比簇的边界层恢复了节点像；关联论证恢复了线性分量。在此过程中，我们修复了Andreotti对光滑曲线Torelli定理证明的经典处理中的技术缺陷。
英文摘要
Let $C$ be a connected nodal curve of genus $g$ over an algebraically closed field $k$ of characteristic zero. Let \[ A(C)=\Big(J(C), {\overline{P^{g-1}_C}}, Θ(C) \Big) \] be the stable semi-abelic pair (in the sense of Alexeev) associated with its canonical compactified Jacobian $\overline{P^{g-1}_C}$ in degree $g-1$. The canonical triple of $C$ consists of the closure of its canonical image together with two sets of distinguished points encoding the information of the images of the nonseparating nodes and of the branch data of the canonical map. We prove a functorial reconstruction: given connected nodal curves $C$ and $C'$ over $k$, every isomorphism $A(C)\cong A(C')$ of stable semi-abelic pairs canonically induces a projectivity between the corresponding canonical triples of $C$ and $C'$. Passing to isomorphism classes, this recovers the reconstruction statement in the compactified Torelli theorem of Caporaso-Viviani. Our proof extends Andreotti's Gauss-map strategy. The branch locus of the projection from the normalization of the Gauss graph recovers the nonlinear canonical components and branch data by projective biduality; boundary strata of the compactified Jacobian recover the node-images; and incidence arguments recover the linear components. Along the way, we repair technical gaps in the classical treatments of Andreotti's proof of Torelli's theorem for smooth curves.
