曲线上向量丛的 Quot 概形的 Torelli 定理

中文摘要 AI 辅助

设 $E$ 是光滑射影曲线 $C$ 上的一个向量丛，定义 Quot 概形 $Q_d(E,C)$，它参数化 $E$ 的具有长度 $d$ 挠商（torsion quotients）的子层。若 $E_i$ 是光滑射影曲线 $C_i$ 上的秩 $r\geq 2$ 的向量丛（$i=1,2$），则保持射影丛结构的 $\u0070\u0072\u006f\u006a\u0065\u0063\u0074\u0069\u0076\u0065\u0020\u0062\u0075\u006e\u0064\u006c\u0065\u0020\u0073\u0074\u0072\u0075\u0063\u0074\u0075\u0072\u0065\u0073$ 之间的同构 $\u0070\u0072\u006f\u006a\u0065\u0063\u0074\u0069\u0076\u0065\u0020\u0062\u0075\u006e\u0064\u006c\u0065\u0020\u0073\u0074\u0072\u0075\u0063\u0074\u0075\u0072\u0065\u0073$ 诱导出 $Q_d(E_i,C_i)$ 之间的同构，称为自然同构。我们证明：对于 $d\geq 2$，$Q_d(E_i,C_i)$ 之间的所有同构都是自然的，除了 $Q_2(\mathcal{O}_C^{\oplus r},C)$ 上的一个非自然对合（non-natural involution）。这给出了 $Q_d(E,C)$ 的自同构群的完整描述，推广了 Biswas-Dhillon-Hurtubise 和 Gangopadhyay 的先前工作。这也表明可以从 Quot 概形重建曲线，即一个 Torelli 型定理。作为证明的关键步骤，我们证明光滑射影曲线的对称幂之间保持大对角线（big diagonals）的任何同构都是自然的，这一结果本身也很有趣。

英文摘要

For a vector bundle $E$ on a smooth projective curve $C$, one defines the Quot scheme $Q_d(E,C)$ parametrizing subsheaves of $E$ having length $d$ torsion quotients. If $E_i$ is a vector bundle of rank $r\geq 2$ on a smooth projective curve $C_i$ for $i=1,2$, isomorphisms between $\mathbb{P}_{C_i}(E_i)$ preserving the projective bundle structures induce isomorphisms of $Q_d(E_i,C_i)$, called natural isomorphisms. We show that for $d\geq 2$ all isomorphisms between $Q_d(E_i,C_i)$ are natural, except for a non-natural involution of $Q_2(\mathcal{O}_C^{\oplus r},C)$. This gives a complete description of the automorphism group of $Q_d(E,C)$, generalizing previous works of Biswas-Dhillon-Hurtubise and Gangopadhyay. This also shows that one can reconstruct the curve from the Quot scheme, a Torelli-type theorem. As a key step in our proof, we show that any isomorphism between symmetric powers of smooth projective curves preserving big diagonals is natural, which is interesting in its own right.

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20 pages. Comments are welcome