Topological Structures of Moduli Spaces of Curves and Anabelian Geometry in Positive Characteristic

TL;DR

This paper explores how the topological structures of moduli spaces of curves in positive characteristic can be understood through anabelian geometry, proposing new conjectures and proving them for genus 0 cases.

Contribution

It introduces new anabelian-geometric conjectures relating tame fundamental groups to moduli spaces of curves in positive characteristic, extending Tamagawa's work.

Findings

Conjectures relate fundamental groups to moduli spaces in characteristic p.

Proved conjectures for genus 0 curves.

Links topological structures with anabelian geometry in positive characteristic.

Abstract

In the present paper, we study a new kind of anabelian phenomenon concerning the smooth pointed stable curves in positive characteristic. It shows that the topological structures of moduli spaces of curves can be understood from the viewpoint of anabelian geometry. We formulate some new anabelian-geometric conjectures relating the tame fundamental groups of curves over algebraically closed fields of characteristic $p>0$ to the moduli spaces of curves. These conjectures are generalized versions of the weak Isom-version of the Grothendieck conjecture for curves over algebraically closed fields of characteristic $p>0$ which was formulated by Tamagawa. Moreover, we prove that the conjectures hold for certain points lying in the moduli space of curves of genus $0$.