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?


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

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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.