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| Format: | Preprint |
| Veröffentlicht: |
2025
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| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2509.11227 |
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| _version_ | 1866912816384966656 |
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| author | Choi, Youngook Iliev, Hristo Kim, Seonja |
| author_facet | Choi, Youngook Iliev, Hristo Kim, Seonja |
| contents | In this note, we show that for a smooth algebraic variety $Y$ and a smooth $m$-section $X$ of the $\mathbb{P}^1$-bundle \[ f : \mathbb{P}(\mathcal{O}_Y \oplus \mathcal{O}_Y(E)) \longrightarrow Y, \] where $E$ is an effective divisor on $Y$ satisfying $H^1(Y, \mathcal{O}_Y(kE)) = 0$ for all $k = 1, \ldots, m-1$, the Tschirnhausen module of the induced covering $
f|_X : X \longrightarrow Y $ is completely decomposable. We then apply it to coverings of curves arising in such a way. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_11227 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Decomposition of the Tschirnhausen module for coverings on decomposable $\mathbb{P}^1$-bundles Choi, Youngook Iliev, Hristo Kim, Seonja Algebraic Geometry In this note, we show that for a smooth algebraic variety $Y$ and a smooth $m$-section $X$ of the $\mathbb{P}^1$-bundle \[ f : \mathbb{P}(\mathcal{O}_Y \oplus \mathcal{O}_Y(E)) \longrightarrow Y, \] where $E$ is an effective divisor on $Y$ satisfying $H^1(Y, \mathcal{O}_Y(kE)) = 0$ for all $k = 1, \ldots, m-1$, the Tschirnhausen module of the induced covering $ f|_X : X \longrightarrow Y $ is completely decomposable. We then apply it to coverings of curves arising in such a way. |
| title | Decomposition of the Tschirnhausen module for coverings on decomposable $\mathbb{P}^1$-bundles |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2509.11227 |