Let $\mathcal{M}_{0,n}$ be the complement of the boundary of the MumfordKnudsen compactification of the moduli space of genus zero, npointed curves.
Is $Pic(\mathcal{M}_{0,n})$ trivial?
Let $\mathcal{M}_{0,n}$ be the complement of the boundary of the MumfordKnudsen compactification of the moduli space of genus zero, npointed curves. Is $Pic(\mathcal{M}_{0,n})$ trivial? 


Yes. By fixing the three points $\{0,1,\infty\}$ one sees that $M_{0,n}$ is isomorphic to an open subscheme of $\mathbb{A}^{n3}$ which has trivial Picard group. Since it is smooth, the Picard group of any open subscheme is also trivial. 

