On the Admissible Fundamental Groups of Curves over Algebraically Closed Fields of Characteristic $p>0$

Abstract

In the present paper, we study the anabelian geometry of pointed stable curves over algebraically closed fields of positive characteristic. We prove that the semi-graph of anabelioids of PSC-type arising from a pointed stable curve over an algebraically closed field of positive characteristic can be reconstructed group-theoretically from its fundamental group. This result may be regarded as a version of the combinatorial Grothendieck conjecture in positive characteristic. As an application, we prove that if a pointed stable curve over an algebraic closure of a finite field satisfies certain conditions, then the isomorphism class of the admissible fundamental group of the pointed stable curve completely determines the isomorphism class of the pointed stable curve as a scheme. This result generalizes a result of A. Tamagawa to the case of (possibly singular) pointed stable curves.