Fourier-Mukai transform for fine compactified Prym varieties
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866910345783672832 |
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| author | Franco, Emilio Hanson, Robert Ruano, João |
| author_facet | Franco, Emilio Hanson, Robert Ruano, João |
| contents | Consider a finite covering $β: C \to X$ of a smooth projective curve $X$ by a reduced, projective, planar curve $C$. Associated to two general polarizations on $C$, $q$ and $q'$, one can construct the corresponding compactified Prym varieties $\overline{\mathrm{P}}_β(q)$ and $\overline{\mathrm{P}}_β(q')$. Consider $Γ$ to be the group of line bundles whose torsion coincides with the order of $β$. In this article we construct a Fourier-Mukai transform between the derived categories of $\overline{\mathrm{P}}_β(q)$ and the $Γ$-equivariant derived category of $\overline{\mathrm{P}}_β(q')$. Hence, we obtain a derived equivalence between the $\mathrm{SL}(n,\mathbb{C})$-Hitchin fibre and its associated $\mathrm{PGL}(n,\mathbb{C})$-Hitchin fibre for a dense class of singular spectral curves. Our work then provides the extension of the Fourier-Mukai transform constructed by Arinkin and Melo-Rapagnetta-Viviani, which corresponds to autoduality of $\mathrm{GL}(n,\mathbb{C})$-Hitchin fibres in this class of singular spectral curves. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_12349 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Fourier-Mukai transform for fine compactified Prym varieties Franco, Emilio Hanson, Robert Ruano, João Algebraic Geometry Consider a finite covering $β: C \to X$ of a smooth projective curve $X$ by a reduced, projective, planar curve $C$. Associated to two general polarizations on $C$, $q$ and $q'$, one can construct the corresponding compactified Prym varieties $\overline{\mathrm{P}}_β(q)$ and $\overline{\mathrm{P}}_β(q')$. Consider $Γ$ to be the group of line bundles whose torsion coincides with the order of $β$. In this article we construct a Fourier-Mukai transform between the derived categories of $\overline{\mathrm{P}}_β(q)$ and the $Γ$-equivariant derived category of $\overline{\mathrm{P}}_β(q')$. Hence, we obtain a derived equivalence between the $\mathrm{SL}(n,\mathbb{C})$-Hitchin fibre and its associated $\mathrm{PGL}(n,\mathbb{C})$-Hitchin fibre for a dense class of singular spectral curves. Our work then provides the extension of the Fourier-Mukai transform constructed by Arinkin and Melo-Rapagnetta-Viviani, which corresponds to autoduality of $\mathrm{GL}(n,\mathbb{C})$-Hitchin fibres in this class of singular spectral curves. |
| title | Fourier-Mukai transform for fine compactified Prym varieties |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2206.12349 |