Timeline for $\mathbb CP^k$ bundles over $\mathbb CP^n$ are projectivisations of vector bundles. Any reference?
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| S Dec 10, 2019 at 18:02 | history | bounty ended | CommunityBot | ||
| S Dec 10, 2019 at 18:02 | history | notice removed | CommunityBot | ||
| Dec 10, 2019 at 13:15 | vote | accept | aglearner | ||
| Dec 10, 2019 at 13:14 | answer | added | aglearner | timeline score: 0 | |
| Dec 9, 2019 at 9:21 | answer | added | aglearner | timeline score: 1 | |
| Dec 6, 2019 at 21:30 | history | edited | Ali Taghavi |
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| Dec 5, 2019 at 23:18 | comment | added | gcousin | If I remember well there are discussions on this theme in Grothendieck-Serre correspondence. | |
| Dec 2, 2019 at 18:22 | history | edited | aglearner | CC BY-SA 4.0 |
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| Dec 2, 2019 at 17:57 | comment | added | aglearner | Thanks for clarifying this point Donu! Maybe you have encountered such a statement about $\mathbb CP^k$-bundles over $\mathbb CP^n$ somewhere? | |
| Dec 2, 2019 at 17:36 | comment | added | Donu Arapura | @aglearner There is no contradiction. A projective space bundle, which is locally trivial for the Zariski topology is not the same as a bundle in the analytic or etale topologies. You and the references probably use the latter,and Hartshorne the former. | |
| S Dec 2, 2019 at 16:55 | history | bounty started | aglearner | ||
| S Dec 2, 2019 at 16:55 | history | notice added | aglearner | Authoritative reference needed | |
| Dec 2, 2019 at 16:51 | comment | added | aglearner | Dear Delvin, now I am really puzzled. I look in the paper of Beauville, arxiv.org/pdf/1507.02476.pdf and on the top of Page 20 it is written there : "Thus we associate to each $P^1$-bundle $p : P \to V$ a class $[p]$ in $H^2(V, \mathbb G_m)$, and this class is trivial if and only if $p$ is a projective bundle". I deduce from this that there exist bundles for which this class is non-trivial and so they are not projective. I guess I am missing something here... | |
| Dec 2, 2019 at 16:04 | comment | added | Devlin Mallory | Since you mention Hartshorne's book, I'll just add that this is the content of Exercise II.7.10(c), which states that projective bundles over any regular noetherian scheme arise as the projectivization of a vector bundle on the base. (There is also a hint in the exercise on how to show this.) | |
| Dec 1, 2019 at 17:07 | history | edited | aglearner | CC BY-SA 4.0 |
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| Dec 1, 2019 at 11:24 | comment | added | aglearner | Daniel, thanks for the link. I want a reference for the equivalent statement, since I was learning complex algebraic geometry in rather Griffiths-Harris style, than Hartshorne's. And then, this statement is out of the scope even of Hartshorne's book as far as I can see ... | |
| Dec 1, 2019 at 1:11 | comment | added | Daniel Litt | See e.g. this question for a proof. mathoverflow.net/questions/75774/… I'm not sure why you want a reference for the equivalent statement, but Grothendieck's "Dix Exposes" will certainly suffice as a reference for the equivalence. | |
| Nov 30, 2019 at 20:45 | history | edited | aglearner | CC BY-SA 4.0 |
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| Nov 30, 2019 at 19:08 | history | edited | aglearner | CC BY-SA 4.0 |
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| Nov 30, 2019 at 18:54 | answer | added | Arun Debray | timeline score: 8 | |
| Nov 30, 2019 at 17:29 | history | edited | aglearner | CC BY-SA 4.0 |
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| Nov 30, 2019 at 17:21 | comment | added | A.Garcia | The Brauer group of $P^n(\mathbb{C})$ equals the unramified Brauer group of $\mathbb{C}(t_1,\dots,t_n)$. For every field $k$, one has $Br_{nr}(k(t))=Br_{nr}(k)$, and for an algebraically closed field the unramified Brauer group vanishes (being a subgroup of the Brauer group); see Gille-Szamuely's book "Central simple algebras and Galois cohomology". Using the short exact sequence for $1\to G_m \to GL_n \to PGL_n\to 1$, and passing to etale cohomology, one deduces your statement on projective bundles. | |
| Nov 30, 2019 at 16:54 | history | asked | aglearner | CC BY-SA 4.0 |