Structure of connected nested automorphism groups
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866908888332238848 |
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| author | Perepechko, Alexander |
| author_facet | Perepechko, Alexander |
| contents | In this article, we describe the maximal unipotent subgroups of $\mathrm{Aut}(X)$, where $X$ is an affine algebraic variety. Every subgroup of this type has a structure analogous to that of the group of triangular automorphisms of $\mathbb{A}^n$. In particular, it is nested, that is, a countable increasing union of algebraic subgroups.
We show that a subgroup $G\subset\mathrm{Aut}(X)$ consisting of unipotent elements is closed if and only if it is nested. This implies that a connected nested subgroup of $\mathrm{Aut}(X)$ is closed, thus answering a question posed by Kraft and Zaidenberg (2022).
We also extend the recent description of maximal commutative unipotent subgroups of $\mathrm{Aut}(X)$ due to Regeta and van Santen (2024), by providing a direct construction of such subgroups within our approach. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_08359 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Structure of connected nested automorphism groups Perepechko, Alexander Algebraic Geometry 14J50, 14R20 (Primary) 14L40, 22E65 (Secondary) In this article, we describe the maximal unipotent subgroups of $\mathrm{Aut}(X)$, where $X$ is an affine algebraic variety. Every subgroup of this type has a structure analogous to that of the group of triangular automorphisms of $\mathbb{A}^n$. In particular, it is nested, that is, a countable increasing union of algebraic subgroups. We show that a subgroup $G\subset\mathrm{Aut}(X)$ consisting of unipotent elements is closed if and only if it is nested. This implies that a connected nested subgroup of $\mathrm{Aut}(X)$ is closed, thus answering a question posed by Kraft and Zaidenberg (2022). We also extend the recent description of maximal commutative unipotent subgroups of $\mathrm{Aut}(X)$ due to Regeta and van Santen (2024), by providing a direct construction of such subgroups within our approach. |
| title | Structure of connected nested automorphism groups |
| topic | Algebraic Geometry 14J50, 14R20 (Primary) 14L40, 22E65 (Secondary) |
| url | https://arxiv.org/abs/2312.08359 |