Timeline for Picard Group of Projective Bundle over an Integral scheme
Current License: CC BY-SA 3.0
10 events
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| Apr 11, 2011 at 12:12 | history | edited | Georges Elencwajg | CC BY-SA 3.0 |
added an "edit"
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| Apr 11, 2011 at 11:45 | comment | added | Georges Elencwajg | Dear HNuer, you are absolutely right . I'll edit my answer to acknowledge this. | |
| Apr 11, 2011 at 7:07 | comment | added | HNuer | So certainly every line bundle can be written this way, and injectivity follows from the properties of $\mathcal O_{\mathbb P(\mathcal E)}$ and the projection formula for locally free sheaves. | |
| Apr 11, 2011 at 7:06 | comment | added | HNuer | , the line bundle $\mathcal N$ is on $Y$, $\mathcal M\cong \mathcal O_{\mathbb P(\mathcal E)}(m)\otimes \pi^* \mathcal N$ | |
| Apr 11, 2011 at 7:04 | comment | added | HNuer | I understand your question now. In my answer I just showed why if we take the line bundle $M$ on $\mathbb P(\mathcal E)$ it is isomorphic on each fiber to the same $\mathcal O_{\mathbb P^n}(m)$ and that this $m$ doesn't change. Once we have that, we in the situation of Hartshorne exercise III,12.4 with a flat projective morphism $\pi:\mathbb P(\mathcal E)\rightarrow Y$ and two invertible sheaves $\mathcal M$ and $\mathcal O_{\mathbb P(\mathcal E)}(m)$ which are isormorphis on each fiber. By that exercise (which is essentially Grauert's theorem), we get a line bundle $\mathcal N$ with | |
| Apr 10, 2011 at 21:53 | comment | added | Georges Elencwajg | There is a misunderstanding here. My claim is that the formula $Pic(\mathbb P(\mathcal E) )=p^\ast Pic(Y)\oplus \mathbb Z \xi $ is proved above only under the local factoriality hypothesis. In particular if a bundle on $\mathbb P(\mathcal E)$ is trivial on all fibers $p^{-1}(y)$, is it of the form $p^{\ast} \mathcal L$ for some line bundle $ \mathcal L$ on $Y$, if nothing is assumed on the singularities of $Y$? This question has not been addressed in the comments nor in your answer and since in the original question you seem to claim that the formula is true, could you plese explain why ? | |
| Apr 9, 2011 at 17:36 | comment | added | HNuer | I don't yet know too much above the Chow ring, so I didn't understand your answer so much. Either way, I am in fact trying to prove this for a general integral scheme, and it seems to be true there, so I'm not assuming anything about the singularities of my scheme. | |
| Apr 9, 2011 at 11:09 | history | edited | Georges Elencwajg | CC BY-SA 3.0 |
added "of local factoriality"
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| Apr 8, 2011 at 19:39 | history | edited | Georges Elencwajg | CC BY-SA 3.0 |
Added condition "Y regular" in first sentence and added that second formula didn't need that hypothesis.
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| Apr 8, 2011 at 19:09 | history | answered | Georges Elencwajg | CC BY-SA 3.0 |