A Birational Anabelian Reconstruction Theorem for Curves over Algebraically Closed Fields in Arbitrary Characteristic

The aim of Bogomolov’s programme is to prove birational anabelian conjectures for function fields $K\mathrm{\mid }k$ of varieties of dimension $\ge 2$ over algebraically closed fields. The present article is concerned with the 1-dimensional case. While it is impossible to recover $K\mathrm{\mid }k$ from its absolute Galois group alone, we prove that it can be recovered from the pair $(\mathrm{A}\mathrm{u}\mathrm{t}(\bar{K}\mathrm{\mid }k),\mathrm{A}\mathrm{u}\mathrm{t}(\bar{K}\mathrm{\mid }K))$, consisting of the absolute Galois group of $K$ and the larger group of field automorphisms fixing only the base field.