On the decomposition group of a nonsingular plane cubic by a log Calabi-Yau geometrical perspective
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866914688169672704 |
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| author | da Silva, Eduardo Alves |
| author_facet | da Silva, Eduardo Alves |
| contents | This paper aims to study the decomposition group of a nonsingular plane cubic under the light of the log Calabi-Yau geometry. Using this approach we prove that an appropriate algorithm of the Sarkisov Program in dimension 2 applied to an element of this group is automatically volume preserving. From this, we deduce some properties of the (volume preserving) Sarkisov factorization of its elements. We also negatively answer a question posed by Blanc, Pan and Vust asking whether the canonical complex of a nonsingular plane cubic is split. Within a similar context em dimension 3, we exhibit in detail an interesting counterexample for a possible generalization of a theorem by Pan in which there exists a Sarkisov factorization obtained algorithmically that is not volume preserving. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_13968 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the decomposition group of a nonsingular plane cubic by a log Calabi-Yau geometrical perspective da Silva, Eduardo Alves Algebraic Geometry 14E05, 14E07 (Primary) 14E30 (Secondary) This paper aims to study the decomposition group of a nonsingular plane cubic under the light of the log Calabi-Yau geometry. Using this approach we prove that an appropriate algorithm of the Sarkisov Program in dimension 2 applied to an element of this group is automatically volume preserving. From this, we deduce some properties of the (volume preserving) Sarkisov factorization of its elements. We also negatively answer a question posed by Blanc, Pan and Vust asking whether the canonical complex of a nonsingular plane cubic is split. Within a similar context em dimension 3, we exhibit in detail an interesting counterexample for a possible generalization of a theorem by Pan in which there exists a Sarkisov factorization obtained algorithmically that is not volume preserving. |
| title | On the decomposition group of a nonsingular plane cubic by a log Calabi-Yau geometrical perspective |
| topic | Algebraic Geometry 14E05, 14E07 (Primary) 14E30 (Secondary) |
| url | https://arxiv.org/abs/2402.13968 |