Derived categories of curves of genus one and torsors over abelian varieties
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| Format: | Preprint |
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2022
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| _version_ | 1866915378123243520 |
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| author | Ramachandran, Niranjan Rosenberg, Jonathan |
| author_facet | Ramachandran, Niranjan Rosenberg, Jonathan |
| contents | Suppose $C$ is a smooth projective curve of genus 1 over a perfect field $F$, and $E$ is its Jacobian. In the case that $C$ has no $F$-rational points, so that $C$ and $E$ are not isomorphic, $C$ is an $E$-torsor with a class $δ(C)\in H^1(\text{Gal}(\bar F/F), E(\bar F))$. Then $δ(C)$ determines a class $β\in \text{Br}(E)/\text{Br}(F)$ and there is a Fourier-Mukai equivalence of derived categories of (twisted) coherent sheaves $\mathcal D(C) \xrightarrow{\cong} \mathcal D(E, β^{-1})$. We generalize this result to higher dimensions; namely, we prove it also for torsors over abelian varieties. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2212_14497 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Derived categories of curves of genus one and torsors over abelian varieties Ramachandran, Niranjan Rosenberg, Jonathan Algebraic Geometry High Energy Physics - Theory 14F08 (Primary) 14H52, 18G80, 14F22, 81T30, 81T35 (Secondary) Suppose $C$ is a smooth projective curve of genus 1 over a perfect field $F$, and $E$ is its Jacobian. In the case that $C$ has no $F$-rational points, so that $C$ and $E$ are not isomorphic, $C$ is an $E$-torsor with a class $δ(C)\in H^1(\text{Gal}(\bar F/F), E(\bar F))$. Then $δ(C)$ determines a class $β\in \text{Br}(E)/\text{Br}(F)$ and there is a Fourier-Mukai equivalence of derived categories of (twisted) coherent sheaves $\mathcal D(C) \xrightarrow{\cong} \mathcal D(E, β^{-1})$. We generalize this result to higher dimensions; namely, we prove it also for torsors over abelian varieties. |
| title | Derived categories of curves of genus one and torsors over abelian varieties |
| topic | Algebraic Geometry High Energy Physics - Theory 14F08 (Primary) 14H52, 18G80, 14F22, 81T30, 81T35 (Secondary) |
| url | https://arxiv.org/abs/2212.14497 |