Group Theoretical Characterizations of Rationality
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866914947928162304 |
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| author | Regeta, Andriy Urech, Christian van Santen, Immanuel |
| author_facet | Regeta, Andriy Urech, Christian van Santen, Immanuel |
| contents | Let X be an irreducible variety and Bir(X) its group of birational transformations. We show that the group structure of Bir(X) determines whether X is rational and whether X is ruled.
Additionally, we prove that any Borel subgroup of Bir(X) has derived length at most twice the dimension of X, with equality occurring if and only if X is rational and the Borel subgroup is standard. We also provide examples of non-standard Borel subgroups of Bir(P^n) and Aut(A^n), thereby resolving conjectures by Popov and Furter-Poloni. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_07864 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Group Theoretical Characterizations of Rationality Regeta, Andriy Urech, Christian van Santen, Immanuel Algebraic Geometry 14E07, 14E08, 14L30 Let X be an irreducible variety and Bir(X) its group of birational transformations. We show that the group structure of Bir(X) determines whether X is rational and whether X is ruled. Additionally, we prove that any Borel subgroup of Bir(X) has derived length at most twice the dimension of X, with equality occurring if and only if X is rational and the Borel subgroup is standard. We also provide examples of non-standard Borel subgroups of Bir(P^n) and Aut(A^n), thereby resolving conjectures by Popov and Furter-Poloni. |
| title | Group Theoretical Characterizations of Rationality |
| topic | Algebraic Geometry 14E07, 14E08, 14L30 |
| url | https://arxiv.org/abs/2409.07864 |