Fourier-Mukai transforms and normalisation of nodal curves
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866914513948770304 |
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| author | Franco, Emilio Hanson, Robert Horn, Johannes Oliveira, André |
| author_facet | Franco, Emilio Hanson, Robert Horn, Johannes Oliveira, André |
| contents | We study Arinkin's Poincaré sheaf $\mathcal{P}_C$ on the singular locus of $\overline{\mathsf{Jac}}_C$, the compactified Jacobian of rank one torsion-free sheaves on an integral nodal projective curve $C$. Each stratum of the singular locus $\mathsf{Sing}(\overline{\mathsf{Jac}}_C)$ is indexed by a partial normalisation $Σ\to C$. We prove that the Poincaré sheaf $\mathcal{P}_C$ restricted to each stratum can be expressed through the Poincaré sheaf $\mathcal{P}_Σ$, obtaining a relation between Fourier-Mukai transforms associated to $\mathcal{P}_C$ and $\mathcal{P}_Σ$. Our approach uses an intermediate geometry: the moduli space of parabolic modules of Bhosle and Cook, to intertwine sheaf data over the two curves. In a sequel, our formulae are used to study mirror symmetry in singular loci of Hitchin systems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_11860 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Fourier-Mukai transforms and normalisation of nodal curves Franco, Emilio Hanson, Robert Horn, Johannes Oliveira, André Algebraic Geometry We study Arinkin's Poincaré sheaf $\mathcal{P}_C$ on the singular locus of $\overline{\mathsf{Jac}}_C$, the compactified Jacobian of rank one torsion-free sheaves on an integral nodal projective curve $C$. Each stratum of the singular locus $\mathsf{Sing}(\overline{\mathsf{Jac}}_C)$ is indexed by a partial normalisation $Σ\to C$. We prove that the Poincaré sheaf $\mathcal{P}_C$ restricted to each stratum can be expressed through the Poincaré sheaf $\mathcal{P}_Σ$, obtaining a relation between Fourier-Mukai transforms associated to $\mathcal{P}_C$ and $\mathcal{P}_Σ$. Our approach uses an intermediate geometry: the moduli space of parabolic modules of Bhosle and Cook, to intertwine sheaf data over the two curves. In a sequel, our formulae are used to study mirror symmetry in singular loci of Hitchin systems. |
| title | Fourier-Mukai transforms and normalisation of nodal curves |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2405.11860 |