Remarks on hypersurfaces in $\mathbf{P}^1\times Z$
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866913169845256192 |
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| author | Denisi, Francesco Antonio Laface, Antonio |
| author_facet | Denisi, Francesco Antonio Laface, Antonio |
| contents | We study the birational geometry of hypersurfaces in projective varieties of the form $\mathbf{P}^1\times Z$, where $Z$ satisfies mild assumptions. Building on recent results of Herrera--Laface--Ugaglia, we study their Cox rings (when finitely generated) in terms of the Cox ring of $Z$. In particular, we obtain a complete picture when $Z$ is a Fano variety of dimension at least 3, and with class group of rank 1. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_29515 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Remarks on hypersurfaces in $\mathbf{P}^1\times Z$ Denisi, Francesco Antonio Laface, Antonio Algebraic Geometry Commutative Algebra We study the birational geometry of hypersurfaces in projective varieties of the form $\mathbf{P}^1\times Z$, where $Z$ satisfies mild assumptions. Building on recent results of Herrera--Laface--Ugaglia, we study their Cox rings (when finitely generated) in terms of the Cox ring of $Z$. In particular, we obtain a complete picture when $Z$ is a Fano variety of dimension at least 3, and with class group of rank 1. |
| title | Remarks on hypersurfaces in $\mathbf{P}^1\times Z$ |
| topic | Algebraic Geometry Commutative Algebra |
| url | https://arxiv.org/abs/2605.29515 |