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| Natura: | Preprint |
| Pubblicazione: |
2023
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| Accesso online: | https://arxiv.org/abs/2310.02413 |
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| _version_ | 1866909704132755456 |
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| author | Banerjee, Kuntal Rayan, Steven |
| author_facet | Banerjee, Kuntal Rayan, Steven |
| contents | We explore a strong categorical correspondence between isomorphism classes of sheaves of arbitrary rank on a given algebraic curve and twisted pairs on another algebraic curve, mostly from a linear-algebraic standpoint. In a particular application, we realize a generic elliptic curve as a spectral cover of the complex projective line $\mathbb{P}^1$ and then construct examples of cyclic pairs and co-Higgs bundles over $\mathbb{P}^1$. By appealing to a composite push-pull projection formula, we conjecture an iterated version of spectral correspondence. We prove this conjecture for a particular class of spectral covers of $\mathbb {P}^1$ through Galois-theoretic arguments. The proof relies upon a classification of Galois groups into primitive and imprimitive types. In this context, we revisit a classical theorem of Ritt. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_02413 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A generalized spectral correspondence Banerjee, Kuntal Rayan, Steven Algebraic Geometry Representation Theory 14D20, 14H60, 20B35 We explore a strong categorical correspondence between isomorphism classes of sheaves of arbitrary rank on a given algebraic curve and twisted pairs on another algebraic curve, mostly from a linear-algebraic standpoint. In a particular application, we realize a generic elliptic curve as a spectral cover of the complex projective line $\mathbb{P}^1$ and then construct examples of cyclic pairs and co-Higgs bundles over $\mathbb{P}^1$. By appealing to a composite push-pull projection formula, we conjecture an iterated version of spectral correspondence. We prove this conjecture for a particular class of spectral covers of $\mathbb {P}^1$ through Galois-theoretic arguments. The proof relies upon a classification of Galois groups into primitive and imprimitive types. In this context, we revisit a classical theorem of Ritt. |
| title | A generalized spectral correspondence |
| topic | Algebraic Geometry Representation Theory 14D20, 14H60, 20B35 |
| url | https://arxiv.org/abs/2310.02413 |