Birational automorphism groups of Severi-Brauer surfaces over the field of rational numbers
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arXiv
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| Format: | Preprint |
| Published: |
2022
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| _version_ | 1866908324786601984 |
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| author | Vikulova, Anastasia V. |
| author_facet | Vikulova, Anastasia V. |
| contents | We prove that the only non-trivial finite subgroups of birational automorphism group of non-trivial Severi--Brauer surfaces over the field of rational numbers are~$\mathbb{Z}/3\mathbb{Z}$ and $(\mathbb{Z}/3\mathbb{Z})^2.$ Moreover, we show that $(\mathbb{Z}/3\mathbb{Z})^2$ is contained in $\mathrm{Bir}(V)$ for any Severi--Brauer surface $V$ over a field of characteristic different from $2$ and $3$, and $(\mathbb{Z}/3\mathbb{Z})^3$ is contained in $\mathrm{Bir}(V)$ for any Severi--Brauer surface~$V$ over a field of characteristic different from $2$ and $3$ which contains a non-trivial cube root of unity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_11456 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Birational automorphism groups of Severi-Brauer surfaces over the field of rational numbers Vikulova, Anastasia V. Algebraic Geometry We prove that the only non-trivial finite subgroups of birational automorphism group of non-trivial Severi--Brauer surfaces over the field of rational numbers are~$\mathbb{Z}/3\mathbb{Z}$ and $(\mathbb{Z}/3\mathbb{Z})^2.$ Moreover, we show that $(\mathbb{Z}/3\mathbb{Z})^2$ is contained in $\mathrm{Bir}(V)$ for any Severi--Brauer surface $V$ over a field of characteristic different from $2$ and $3$, and $(\mathbb{Z}/3\mathbb{Z})^3$ is contained in $\mathrm{Bir}(V)$ for any Severi--Brauer surface~$V$ over a field of characteristic different from $2$ and $3$ which contains a non-trivial cube root of unity. |
| title | Birational automorphism groups of Severi-Brauer surfaces over the field of rational numbers |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2211.11456 |