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3
2
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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 $\mathbb{Q}$ ? (Answer known. There is a pun on "locally trivial" in the title.) |
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6
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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 $\mathbb{Q}$. |
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