Let $\mathcal{M}_{0,n}$ be the complement of the boundary of the Mumford-Knudsen compactification of the moduli space of genus zero, n-pointed curves.

Is $Pic(\mathcal{M}_{0,n})$ trivial?

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Yes. By fixing the three points $\{0,1,\infty\}$ one sees that $M_{0,n}$ is isomorphic to an open subscheme of $\mathbb{A}^{n-3}$ which has trivial Picard group. Since it is smooth, the Picard group of any open subscheme is also trivial.

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