On anabelian properties of the moduli spaces of smooth projective curves
Jakob Stix
Anabelian geometry deals with the arithmetical/geometrical content of
the étale fundamental group of a variety.
In particular, it should be possible to describe maps to an anabelian
variety in terms of the
fundamental group alone, as for example this is the case in homotopy
theory with
Eilenberg-MacLane spaces.
The moduli spaces of smooth projective curves are generally believed to
share many anabelian properties.
The talk will explain published results that deal with constant maps and
the problem of extending a
map from some open dense subset, leading to an anabelian and generalized
version of Moret-Bailly's
purity theorem for smooth projective curves.