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

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