Automorphism groups of rigid affine surfaces: the identity component
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866910802761482240 |
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| author | Perepechko, Alexander Zaidenberg, Mikhail |
| author_facet | Perepechko, Alexander Zaidenberg, Mikhail |
| contents | It is known that the identity component of the automorphism group of a projective algebraic variety is an algebraic group. This is not true in general for quasi-projective varieties. In this note we address the question: given an affine algebraic surface $Y$, as to when the identity component ${\rm Aut}^0 (Y)$ of the automorphism group ${\rm Aut} (Y)$ is an algebraic group? We show that this happens if and only if $Y$ admits no effective action of the additive group of the field. In the latter case, ${\rm Aut}^0 (Y)$ is an algebraic torus of rank $\le 2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_09738 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Automorphism groups of rigid affine surfaces: the identity component Perepechko, Alexander Zaidenberg, Mikhail Algebraic Geometry 14J50, 14R20 (primary), 14L30, 05C60 (secondary) It is known that the identity component of the automorphism group of a projective algebraic variety is an algebraic group. This is not true in general for quasi-projective varieties. In this note we address the question: given an affine algebraic surface $Y$, as to when the identity component ${\rm Aut}^0 (Y)$ of the automorphism group ${\rm Aut} (Y)$ is an algebraic group? We show that this happens if and only if $Y$ admits no effective action of the additive group of the field. In the latter case, ${\rm Aut}^0 (Y)$ is an algebraic torus of rank $\le 2$. |
| title | Automorphism groups of rigid affine surfaces: the identity component |
| topic | Algebraic Geometry 14J50, 14R20 (primary), 14L30, 05C60 (secondary) |
| url | https://arxiv.org/abs/2208.09738 |