Classifying compact Riemann surfaces by number of symmetries
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866915130063716352 |
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| author | Reyes-Carocca, Sebastián Speziali, Pietro |
| author_facet | Reyes-Carocca, Sebastián Speziali, Pietro |
| contents | In this article we consider compact Riemann surfaces that are uniquely determined by the property of possessing a group of automorphisms of a prescribed order, strengthening uniqueness results proved by Nakagawa. More precisely, we deal with the cases in which such an order is $3g$ and $3g+3,$ where $g$ is the genus. We prove that if $g$ is odd (respectively $g$ even and $g \not \equiv 2 \mbox{ mod } 3$) then there exists a unique Riemann surface of genus $g$ with a group of automorphisms of order $3g$ (respectively $3g+3$). A similar conclusion can be derived in terms of orientably-regular hypermaps. In addition, we determine the full automorphism group of such Riemann surfaces and provide decompositions of their Jacobians. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2310_07520 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Classifying compact Riemann surfaces by number of symmetries Reyes-Carocca, Sebastián Speziali, Pietro Algebraic Geometry 30F10, 32G15, 14H37, 30F35, 14H30 In this article we consider compact Riemann surfaces that are uniquely determined by the property of possessing a group of automorphisms of a prescribed order, strengthening uniqueness results proved by Nakagawa. More precisely, we deal with the cases in which such an order is $3g$ and $3g+3,$ where $g$ is the genus. We prove that if $g$ is odd (respectively $g$ even and $g \not \equiv 2 \mbox{ mod } 3$) then there exists a unique Riemann surface of genus $g$ with a group of automorphisms of order $3g$ (respectively $3g+3$). A similar conclusion can be derived in terms of orientably-regular hypermaps. In addition, we determine the full automorphism group of such Riemann surfaces and provide decompositions of their Jacobians. |
| title | Classifying compact Riemann surfaces by number of symmetries |
| topic | Algebraic Geometry 30F10, 32G15, 14H37, 30F35, 14H30 |
| url | https://arxiv.org/abs/2310.07520 |