The Cox ring of an embedded variety
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866911703703224320 |
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| author | Herrera, Cristóbal Laface, Antonio Ugaglia, Luca |
| author_facet | Herrera, Cristóbal Laface, Antonio Ugaglia, Luca |
| contents | We compute the Cox ring of an embedded variety $X \subseteq Z$ within a Mori dream space, under the assumption that the pullback map induces an isomorphism at the level of divisor class groups. We show that the Cox ring of $X$ is the intersection of finitely many localizations of a quotient image of the Cox ring of $Z$. As a consequence, we provide an algorithm that terminates if and only if the Cox ring of $X$ is finitely generated, thereby generalizing previous works on the subject. We apply these results to compute the Cox ring of hypersurfaces in smooth projective toric varieties. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_17370 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Cox ring of an embedded variety Herrera, Cristóbal Laface, Antonio Ugaglia, Luca Algebraic Geometry Primary 14M25, Secondary 14C20 We compute the Cox ring of an embedded variety $X \subseteq Z$ within a Mori dream space, under the assumption that the pullback map induces an isomorphism at the level of divisor class groups. We show that the Cox ring of $X$ is the intersection of finitely many localizations of a quotient image of the Cox ring of $Z$. As a consequence, we provide an algorithm that terminates if and only if the Cox ring of $X$ is finitely generated, thereby generalizing previous works on the subject. We apply these results to compute the Cox ring of hypersurfaces in smooth projective toric varieties. |
| title | The Cox ring of an embedded variety |
| topic | Algebraic Geometry Primary 14M25, Secondary 14C20 |
| url | https://arxiv.org/abs/2411.17370 |