On the existence of specialization isomorphisms of admissible fundamental groups in positive characteristic

Let $p$ be a prime number, and let $\overline{\mathcal{M}}_{g,n}$ be the moduli stack over an algebraic closure $\overline{\mathbb{F}}_p$ of the finite field $\mathbb{F}_p$ parameterizing pointed stable curves of type $(g, n), \overline{\mathcal{M}}_{g,n}$ the open substack of $\overline{\mathcal{M}}_{g,n}$ parameterizing smooth pointed stable curves, $\overline{M}_{g,n}$ the coarse moduli space of $\overline{\mathcal{M}}_{g,n}, M_{g,n}$ the coarse moduli space of $\mathcal{M}_{g,n}, q \in \overline{M}_{g,n}$ an arbitrary point, and $\Pi_q$ the admissible fundamental group of a pointed stable curve corresponding to a geometric point over $q$. In the present paper, we prove that, there exists $q_1, q_2 \in \overline{M}_{g,n} \setminus M_{g,n}$ such that $q_1$ is a specialization of $q_2$, that $q_1 \neq q_2$, and that a specialization homomorphism $sp : \Pi_{q_2} \twoheadrightarrow \Pi_{q_1}$ is an isomorphism when $\operatorname{dim}(\overline{M}_{g,n}) \geq 3$.

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This work was supported by JSPS Grant-in-Aid for Young Scientists Grant Numbers 20K14283, and by the Research Institute for Mathematical Sciences (RIMS), an International Joint Usage/ Research Center located in Kyoto University.

Received 22 January 2020

Accepted 25 April 2020

Published 29 August 2022