Timeline for Flat morphisms whose fibers are affine spaces
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
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| Nov 24, 2019 at 12:55 | answer | added | Hanspeter Kraft | timeline score: 4 | |
| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Feb 4, 2013 at 9:09 | comment | added | Damian Rössler | @Lvovski: on second thoughts: is the generic fibre of the morphism $\tilde{X}\to{\bf P}^2$ also a projective space over the function field of ${\bf P}^2$ (I don't think so) ? In order to be a counterexample to Rami's statement, that must hold too. | |
| Feb 3, 2013 at 18:00 | comment | added | Damian Rössler | The basic reference for Severi-Brauer schemes is Grothendieck, "Le groupe de Brauer I" in "Dix exposés sur la cohomologie des schémas" (a collection of essays published by North-Holland). @Lvovski: thank you for the interesting example. | |
| Feb 3, 2013 at 16:51 | comment | added | Calc | @Rami. Thanks a lot for pointing out the other related question. @Damian. What reference do you recommend for Severi-Brauer schemes? | |
| Feb 3, 2013 at 16:19 | comment | added | Serge Lvovski | @Damian: no, this is false even if fibers are actually projective spaces (over an algebraically closed field). A standard example: if $X\subset\mathbb P^4$ is a smooth cubic, then, projecting it from a line $\ell\subset X$ to $\mathbb P^2$, we obtain a morphism $\tilde X\to\mathbb P^2$ (where $\tilde X$ is the blow-up of $X$ at $\ell$). Over a Zariski open subset of $\mathbb P^2$, all its fibers are smooth plane conics (so they are isomorphic to $\mathbb P^1$). If it were a locally trivial bundle in Zariski topology, $X$ would be rational, which is not the case. | |
| Feb 3, 2013 at 14:34 | comment | added | Damian Rössler | ...although: what I said in my comment applies if the fibres are geometrically projective spaces. Maybe what you say works if the fibres are actually projective spaces. | |
| Feb 3, 2013 at 14:27 | comment | added | Damian Rössler | @Rami: In the case of projective spaces, it is only true for étale localisation (not Zariski). Such fibrations are called Severi-Brauer schemes. | |
| Feb 3, 2013 at 2:10 | comment | added | Rami | If instad of the affine space you would have the protective one, than I think that the answer s positive. If you are interested I can try to write a proof | |
| Feb 2, 2013 at 23:07 | comment | added | Rami | Did you check Angelo's answer to mathoverflow.net/questions/58009/non-locally-trivial-an-bundles ? It seems to be very related to your question | |
| Feb 2, 2013 at 22:09 | history | asked | Calc | CC BY-SA 3.0 |