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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2504.15461 |
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| _version_ | 1866916701679910912 |
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| author | Bandman, Tatiana Kunyavskii, Boris Skorobogatov, Alexei N. |
| author_facet | Bandman, Tatiana Kunyavskii, Boris Skorobogatov, Alexei N. |
| contents | We prove that the subvariety of $SL(2)\times SL(2)$ given by the matrix equation $w(X,Y)=α$, where $w$ is a word in two letters, is closely related to an explicit smooth conic bundle over the associated `trace surface' in the 3-dimensional affine space. When $w$ is the commutator word, we show that this variety can be irrational if the ground field $k$ is not algebraically closed, answering a question of Rapinchuk, Benyash-Krivetz, and Chernousov. When $k$ is a number field, it satisfies weak approximation with the Brauer--Manin obstruction. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_15461 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Birational properties of word varieties Bandman, Tatiana Kunyavskii, Boris Skorobogatov, Alexei N. Algebraic Geometry 14E08, 14G05, 14G12, 20G30 We prove that the subvariety of $SL(2)\times SL(2)$ given by the matrix equation $w(X,Y)=α$, where $w$ is a word in two letters, is closely related to an explicit smooth conic bundle over the associated `trace surface' in the 3-dimensional affine space. When $w$ is the commutator word, we show that this variety can be irrational if the ground field $k$ is not algebraically closed, answering a question of Rapinchuk, Benyash-Krivetz, and Chernousov. When $k$ is a number field, it satisfies weak approximation with the Brauer--Manin obstruction. |
| title | Birational properties of word varieties |
| topic | Algebraic Geometry 14E08, 14G05, 14G12, 20G30 |
| url | https://arxiv.org/abs/2504.15461 |