On abelian extensions of finite abelian subgroups of Cremona groups
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911211062296576 |
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| author | Filin, Luka |
| author_facet | Filin, Luka |
| contents | In this note, we study extension properties of finite abelian subgroups of $\mathrm{Bir}(X)$ where $X$ is a rational (or rationally connected) variety of dimension at most $4$. We are guided by the following question: is it true that if a finite group $G$ faithfully acts on a rationally connected variety of dimension $n$, then $G$ can faithfully act on a terminal Fano variety of dimension $n$? Using algebraic methods, we prove that up to dimension $4$, abelian extensions of finite abelian subgroups of the Cremona group coincide with direct products of such subgroups, with one exception. This result implies a positive answer to the above question up to dimension $4$ in the case of finite abelian groups, modulo a conjectural description of finite abelian subgroups of $\mathrm{Bir}(X)$ where $X$ is a rationally connected threefold. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_13200 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On abelian extensions of finite abelian subgroups of Cremona groups Filin, Luka Algebraic Geometry In this note, we study extension properties of finite abelian subgroups of $\mathrm{Bir}(X)$ where $X$ is a rational (or rationally connected) variety of dimension at most $4$. We are guided by the following question: is it true that if a finite group $G$ faithfully acts on a rationally connected variety of dimension $n$, then $G$ can faithfully act on a terminal Fano variety of dimension $n$? Using algebraic methods, we prove that up to dimension $4$, abelian extensions of finite abelian subgroups of the Cremona group coincide with direct products of such subgroups, with one exception. This result implies a positive answer to the above question up to dimension $4$ in the case of finite abelian groups, modulo a conjectural description of finite abelian subgroups of $\mathrm{Bir}(X)$ where $X$ is a rationally connected threefold. |
| title | On abelian extensions of finite abelian subgroups of Cremona groups |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2510.13200 |