Theta functions for singular curves
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866918494944100352 |
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| author | Biswas, Indranil Hurtubise, Jacques |
| author_facet | Biswas, Indranil Hurtubise, Jacques |
| contents | Let $X$ be an irreducible singular Riemann surface, with desingularisation $\widetilde X$. The generalised Jacobian $J(X)$ of $X$ fibers over the Jacobian $J(\widetilde{X})$ of $\widetilde X$, and there is an Abel map $A$ of $\widetilde X$ to $J(X)$, lifting the Abel map to $J(\widetilde X)$. We build a theta function on a compactification of the generalised Jacobian $J(X)$ (giving a section of a suitable positive line bundle). The translation action on $J(X)$ then yields all line bundles of that degree, and the translates of the theta function, restricted to $A(\widetilde X)$, give a ``universal section'' of the line bundles of that degree over $X$. This extends to the singular case a classical result of Riemann. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_11152 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Theta functions for singular curves Biswas, Indranil Hurtubise, Jacques Algebraic Geometry Mathematical Physics 14H60, 14D21, 70G55, 70H06 Let $X$ be an irreducible singular Riemann surface, with desingularisation $\widetilde X$. The generalised Jacobian $J(X)$ of $X$ fibers over the Jacobian $J(\widetilde{X})$ of $\widetilde X$, and there is an Abel map $A$ of $\widetilde X$ to $J(X)$, lifting the Abel map to $J(\widetilde X)$. We build a theta function on a compactification of the generalised Jacobian $J(X)$ (giving a section of a suitable positive line bundle). The translation action on $J(X)$ then yields all line bundles of that degree, and the translates of the theta function, restricted to $A(\widetilde X)$, give a ``universal section'' of the line bundles of that degree over $X$. This extends to the singular case a classical result of Riemann. |
| title | Theta functions for singular curves |
| topic | Algebraic Geometry Mathematical Physics 14H60, 14D21, 70G55, 70H06 |
| url | https://arxiv.org/abs/2605.11152 |