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

Let p be a prime number, and let Mg,n be the moduli stack over an algebraic closure Fp of the finite field Fp parameterizing pointed stable curves of type (g, n), Mg,n the open substack of Mg,n parameterizing smooth pointed stable curves, Mg,n the coarse moduli space of Mg,n, Mg,n the coarse moduli space of Mg,n, q ∈ Mg,n an arbitrary point, and Πq 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 q1, q2 ∈ Mg,n \ Mg,n such that q1 is a specialization of q2, that q1 ̸= q2, and that a specialization homomorphism sp : Πq2 ↠ Πq1 is an isomorphism.