A birational anabelian reconstruction theorem for curves over algebraically closed fields in arbitrary characteristic (Q1617955)

The present article studies birational anabelian reconstruction for function fields $K|k$ of varieties of dimension 1 over algebraically closed fields. This is an extension of the birational anabelian program initiated by \textit{F. A. Bogomolov} [in: Algebraic geometry and analytic geometry. Proceedings of a conference, held in Tokyo, Japan, August 13-17, 1990. Tokyo etc.: Springer-Verlag. 26--52 (1991; Zbl 0789.14021)] which aims at recovering function fields $K|k$ of dimension $> 1$ over algebraically closed fields from their absolute Galois group $G_K$. This cannot be possible in the one-dimensional case since then $G_K$ is profinite free of rank $|k|$ by results of \textit{D. Harbater} [Contemp. Math. 186, 353--369 (1995; Zbl 0858.14013)] and \textit{F. Pop} [Invent. Math. 120, No. 3, 555--578 (1995; Zbl 0842.14017)], containing therefore almost no information about $K$. The authors show however that $K|k$ can be recovered if, in addition to $G_K$, also the larger automorphism group $\mathrm{Aut}(K|k)\supseteq G_K$ fixing only the base field is provided. Also, there is found a Galois-type correspondence for transcendental field extensions and a group-theoretic characterisation of stabiliser subgroups for $\mathrm{PGL}(2,k)$ acting on $\mathbb P^1$.