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| Main Author: | |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2508.09065 |
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| _version_ | 1866916977232052224 |
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| author | Bai, David Zhiyuan |
| author_facet | Bai, David Zhiyuan |
| contents | Let $X\to C$ be an elliptic surface with integral fibers and a section. The Hilbert scheme $X^{[n]}$ fibers over $C^{[n]}$. We construct a commutative group scheme over the entire base $C^{[n]}$ that embeds as an open subscheme of the Hilbert scheme, such that its action on itself extends to the entirety of $X^{[n]}$. We show that the action is $δ$-regular in the sense of Ngô. Using the derived McKay correspondence, we construct an exact autoequivalence of $D^b\operatorname{Coh}(X^{[n]})$ whose kernel is a maximal Cohen-Macaulay sheaf on the fiber product. We show that this Fourier-Mukai transform intertwines with our group action, i.e. theorem of the square holds. We also discuss the case without a section using the theory of Tate-Shafarevich twists. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_09065 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hilbert schemes of elliptic surfaces: group actions and derived categories Bai, David Zhiyuan Algebraic Geometry Let $X\to C$ be an elliptic surface with integral fibers and a section. The Hilbert scheme $X^{[n]}$ fibers over $C^{[n]}$. We construct a commutative group scheme over the entire base $C^{[n]}$ that embeds as an open subscheme of the Hilbert scheme, such that its action on itself extends to the entirety of $X^{[n]}$. We show that the action is $δ$-regular in the sense of Ngô. Using the derived McKay correspondence, we construct an exact autoequivalence of $D^b\operatorname{Coh}(X^{[n]})$ whose kernel is a maximal Cohen-Macaulay sheaf on the fiber product. We show that this Fourier-Mukai transform intertwines with our group action, i.e. theorem of the square holds. We also discuss the case without a section using the theory of Tate-Shafarevich twists. |
| title | Hilbert schemes of elliptic surfaces: group actions and derived categories |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2508.09065 |