Timeline for Birational Automorphisms and infinite divisibility
Current License: CC BY-SA 3.0
17 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| Feb 8, 2013 at 15:57 | history | edited | sfilip | CC BY-SA 3.0 |
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| Feb 7, 2013 at 12:05 | comment | added | Jérémy Blanc | @sfilip: I do not think that the article you mention in your question really proves that Bir(X) embedds as a group into Bir(P^n) for some n. The article of Mella and Polastri only proves that you can extend elements of Bir(X) to Bir(P^n) but not the group structure! For example, no non-affine algebraic group embedds into any Bir(P^n), see Remark 2.21 of arxiv.org/abs/1210.6960 and the reference which is given in it. Here I add the algebraic group structure, so maybe as an abtract group it could a priori embedd, but it is not clear at all how to do it. | |
| Feb 7, 2013 at 11:59 | comment | added | Jérémy Blanc | Exactly, the action only works in dimension $2$. In dimension higher, there are some ideas using valuations but I do not think that there is really a nice description yet. | |
| Feb 6, 2013 at 16:12 | comment | added | sfilip | Unfortunately the Cantat and Lamy results and methods apply only to the planar Cremona group, as far as I understand them. The hyperbolic space comes from the intersection form on the divisors, and that's only available in dimension two. | |
| Feb 6, 2013 at 16:10 | history | edited | sfilip | CC BY-SA 3.0 |
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| Feb 6, 2013 at 8:17 | answer | added | Jérémy Blanc | timeline score: 7 | |
| Feb 6, 2013 at 1:34 | comment | added | Ian Agol | The group $Bir(\mathbb{P}^n)$ is the Cremona group, about which a lot is known: en.wikipedia.org/wiki/Cremona_group Cantat and Lamy proved that it acts as automorphisms of an infinite dimensional hyperbolic space. This might help one understand issues of divisibility. | |
| Feb 6, 2013 at 0:59 | history | edited | sfilip | CC BY-SA 3.0 |
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| Feb 6, 2013 at 0:58 | comment | added | sfilip | @Mariano I was hoping to suggest that "birational action" means a birational map $\mathbb{G}_a\times X \to X$ with the natural compatibility conditions making it an action. This would be one way to make sense of the question, but there might be other reasonable ones. | |
| Feb 5, 2013 at 19:27 | comment | added | Mariano Suárez-Álvarez | @sfilip notice that my comment still applied after your remark: suppose $\mathbb Q$ acts on your variery by birational automorphisms, and let $P$ be a subgroup of $\mathbb R$ such that $\mathbb R=P\oplus\mathbb Q$. Define an action of $\mathbb R$ on the variety so that $(p,q)$ acts just as $q$ acts. Then all elements of $\mathbb R$ do act by birational automorphisms. My point was that you probably want some sort of continuity of the action on the group. | |
| Feb 5, 2013 at 18:52 | answer | added | Dmitri Panov | timeline score: 5 | |
| Feb 5, 2013 at 18:30 | comment | added | sfilip | Apologies for being vague, I've tried to fix the question a bit. Yves, unfortunately I'm ignorant even about the algebraic situation. | |
| Feb 5, 2013 at 18:25 | history | edited | sfilip | CC BY-SA 3.0 |
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| Feb 5, 2013 at 9:26 | comment | added | YCor | Probably it can be asked by: "does not extend to an action of an algebraic group". Btw, the OP asks about birational actions, is the question irrelevant for algebraic actions? | |
| Feb 5, 2013 at 6:17 | comment | added | Mariano Suárez-Álvarez | You want some conditions on the action? As $\mathbb Q$ is a direct summand of the abelian group $\mathbb R$, every action of $\mathbb Q$ (on anything) extends to an action of $\mathbb R$. | |
| Feb 5, 2013 at 1:14 | history | asked | sfilip | CC BY-SA 3.0 |