The tropical Abel--Prym map
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914000174841856 |
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| author | Capobianco, Giusi Len, Yoav |
| author_facet | Capobianco, Giusi Len, Yoav |
| contents | We prove that the tropical Abel--Prym map $Ψ\colon \widetildeΓ\to Prym(\widetildeΓ/Γ)$ associated with a free double cover $π\colon \widetildeΓ\to Γ$ of hyperelliptic metric graphs is harmonic of degree $2$ in accordance with the already established algebraic result. We then prove a partial converse. Contrary to the analogous algebraic result, when the source graph of the double cover is not hyperelliptic, the Abel--Prym map is often not injective.
When the source graph is hyperelliptic, we show that the Abel--Prym graph $Ψ(\widetildeΓ)$ is a hyperelliptic metric graph of genus $g_Γ-1$ whose Jacobian is isomorphic, as pptav, to the Prym variety of the cover. En route,
we count the number of distinct free double covers by hyperelliptic metric graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_06971 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The tropical Abel--Prym map Capobianco, Giusi Len, Yoav Algebraic Geometry 14H40, 14T20 We prove that the tropical Abel--Prym map $Ψ\colon \widetildeΓ\to Prym(\widetildeΓ/Γ)$ associated with a free double cover $π\colon \widetildeΓ\to Γ$ of hyperelliptic metric graphs is harmonic of degree $2$ in accordance with the already established algebraic result. We then prove a partial converse. Contrary to the analogous algebraic result, when the source graph of the double cover is not hyperelliptic, the Abel--Prym map is often not injective. When the source graph is hyperelliptic, we show that the Abel--Prym graph $Ψ(\widetildeΓ)$ is a hyperelliptic metric graph of genus $g_Γ-1$ whose Jacobian is isomorphic, as pptav, to the Prym variety of the cover. En route, we count the number of distinct free double covers by hyperelliptic metric graphs. |
| title | The tropical Abel--Prym map |
| topic | Algebraic Geometry 14H40, 14T20 |
| url | https://arxiv.org/abs/2412.06971 |