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Let $\mathcal{M}_{g,n}$ be the moduli stack over $\mathbb{Q}$ of smooth curves of genus $g$ with $n$ marked points. I've seen in many sources an exact sequence:

$$1\rightarrow\pi_1((\mathcal{M}_{g,n})_{\overline{\mathbb{Q}}})\rightarrow\pi_1(\mathcal{M}_{g,n})\rightarrow\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\rightarrow 1$$

Can someone point to a book/paper where this sequence is proven to be exact?

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V. Zoonekynd, The fundamental group of an algebraic stack, arxiv.org math.AG/0111071

B. Noohi (2004). FUNDAMENTAL GROUPS OF ALGEBRAIC STACKS. Journal of the Institute of Mathematics of Jussieu, 3, pp 69­-103

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  • $\begingroup$ which result in Noohi are you referring to? $\endgroup$ Jul 22 '16 at 6:53
  • $\begingroup$ @rtz Can't find it right now but Cor. 6.6 of Zoonekynd gives what you want. $\endgroup$ Jul 22 '16 at 9:13

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