AI 中文总结
研究\(\mathbb{C}\)上亏格为\(g\)光滑曲线通用族，证明\(g\geq4\)时相对雅可比矩阵下挠子同构于相对皮卡概型连通分支。
AI 中文摘要
我们考虑\(\mathbb{C}\)上亏格为\(g\)的光滑曲线的通用族。证明了对于\(g\geq4\)，相对雅可比矩阵下的每个挠子同构于相对皮卡概型的一个连通分支。
英文摘要
We consider the universal family of smooth genus-$g$ curves over $\mathbb{C}$. We show that for $g\geq4$, every torsor under the relative Jacobian is isomorphic to a connected component of the relative Picard scheme. As a byproduct, we show that $\mathrm{Br}(\mathcal{M}_{3,1})=\mathbb{Z}/2\mathbb{Z}$ over $\mathbb{C}$ and $\mathrm{H}_2(Γ_{3,1},\mathbb{Z})=\mathbb{Z}/2\mathbb{Z}$, pinning down the torsion subgroup in \cite[Theorem 1.2]{zbMATH01991000}.
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