Timeline for Smooth morphism of smooth varieties with fibres isomorphic to an affine space
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
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| Apr 5, 2023 at 19:40 | comment | added | Johan | @DavidRydh Thanks! | |
| Mar 20, 2023 at 13:53 | comment | added | David Rydh | @Johan: A smooth map $f$ with fibers that are all isomorphic to $\mathbb{A}^n$ need not be affine. Jason Starr gives an example $f\colon U\to B$ with $U$ quasi-affine, $B=\mathbb{A}^2$ and $n=2$ in mathoverflow.net/a/136043/40. | |
| May 10, 2021 at 17:48 | vote | accept | Mikhail Borovoi | ||
| May 9, 2021 at 21:21 | comment | added | Johan | Good find! The first thing the paper says is that the paper considers affine morphisms. Is it even clear that a morphism $f$ as in the question is affine? | |
| May 9, 2021 at 19:23 | comment | added | Libli | @P.P : I don't claim that the result isn't true, I am just saying they prove much more. An affine-linear bundle is not just a locally trivial (in the étale topology) morphism with fibers being $\mathbb{A}^n$. As you can read just after definition 2 in the same paper : " every affine-linear bundle is actually locally trivial in the Zariski topology". | |
| May 9, 2021 at 19:19 | comment | added | user211863 | @Libli The main result in the paper of Dubouloz (the theorem right below the passage you quoted) says that if $\Omega^1_{V/X}$ is induced from $X$, then $V$ is affine-linear bundle i.e. it is an étale-locally trivial $\mathbb{A}^2$-bundle on $X$, according to Definition 2. | |
| May 9, 2021 at 19:06 | comment | added | Piotr Achinger | Locally trivial on the source, but not on the target! | |
| May 9, 2021 at 19:04 | comment | added | Libli | Furthermore, if I am not mistake, it is known that a smooth morphism is always locally trivial in the étale topology, isn't it EGA IV, 17.16.3 (ii)? | |
| May 9, 2021 at 19:01 | comment | added | Libli | @PiotrAtchinger I must have a problem with my English today but I read the following : "Summing up, if an $\mathbb{A}^n$-fibration $\pi :V \longrightarrow X$ over an affine scheme $X$ is a Zariski locally trivial $\mathbb{A}^n$-bundle, then its relative cotangent sheaf is induced from $X$. Our main result, which can be summarized as follows, implies in particular that the converse holds for $\mathbb{A}^2$-fibrations over smooth affine schemes". | |
| May 9, 2021 at 18:56 | comment | added | Piotr Achinger | @Libli I'm not an expert, but looking at the theorem in the introduction and Definitions 1 and 2, it seems that the result treats etale-local triviality, no? In any case, Zariski local triviality is even stronger, so gives a better answer to Q1? | |
| May 9, 2021 at 18:48 | comment | added | Libli | I am probably mistaken, but the paper by Dubouloz you mention seems to be interested in local trivilaity in the Zarisi topology, not in the étale topology. | |
| May 9, 2021 at 18:42 | review | First posts | |||
| May 9, 2021 at 20:28 | |||||
| May 9, 2021 at 18:42 | history | answered | user211863 | CC BY-SA 4.0 |