An everywhere locally trivial line bundle - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T07:35:48Z http://mathoverflow.net/feeds/question/10478 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/10478/an-everywhere-locally-trivial-line-bundle An everywhere locally trivial line bundle Chandan Singh Dalawat 2010-01-02T09:24:53Z 2010-01-07T09:38:06Z <p>Is there a variety $X$ over $\mathbb{Q}$ and a line bundle $L$ over $X$ (other than the trivial line bundle $\mathcal{O}_X$ ) such that $L_v$ is the trivial line bundle over $X_v=X\times_{\mathbb{Q}}\mathbb{Q}_v$ for every place $v$ of $\mathbb{Q}$ ?</p> <p>(Answer known. There is a pun on "locally trivial" in the title.)</p> http://mathoverflow.net/questions/10478/an-everywhere-locally-trivial-line-bundle/11027#11027 Answer by Chandan Singh Dalawat for An everywhere locally trivial line bundle Chandan Singh Dalawat 2010-01-07T09:38:06Z 2010-01-07T09:38:06Z <p>The following example was provided to me by Colliot-Thélène some years ago : Let $X$ be the complement in $\mathbb{P}_{1,\mathbb{Q}}$ of the three closed points defined by $x^2=13$, $x^2=17$, $x^2=221$. Then $\operatorname{Pic}(X)=\mathbb{Z}/2\mathbb{Z}$ but $\operatorname{Pic}(X_v)=0$ for every place $v$ of $\mathbb{Q}$.</p>