On topological and combinatorial structures of pointed stable curves over algebraically closed fields of positive characteristic

Yu Yang

Mathematische Nachrichten, 2023, vol. 296, issue 8, 3740-3781

Abstract: In this paper, we study anabelian geometry of curves over algebraically closed fields of positive characteristic. Let X•=(X,DX)$X^{\bullet }=(X, D_{X})$ be a pointed stable curve over an algebraically closed field of characteristic p>0$p>0$ and ΠX•$\Pi _{X^{\bullet }}$ the admissible fundamental group of X•$X^{\bullet }$. We prove that there exists a group‐theoretical algorithm, whose input datum is the admissible fundamental group ΠX•$\Pi _{X^{\bullet }}$, and whose output data are the topological and the combinatorial structures associated with X•$X^{\bullet }$. This result can be regarded as a mono‐anabelian version of the combinatorial Grothendieck conjecture in the positive characteristic. Moreover, by applying this result, we construct clutching maps for moduli spaces of admissible fundamental groups.

Date: 2023
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https://doi.org/10.1002/mana.202100053

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