Reconstruction of higher-dimensional function fields

TL;DR

This paper demonstrates that higher-dimensional function fields over algebraic closures of finite fields can be uniquely identified, up to inseparable extensions, by specific quotients of their pro- Galois groups, advancing understanding in algebraic geometry and Galois theory.

Contribution

It establishes a new Galois-theoretic characterization of higher-dimensional function fields, linking their structure to quotients of their pro- Galois groups.

Findings

Function fields of dimension  are determined by Galois group quotients.

Purely inseparable extensions are the only ambiguities in the identification.

The second term in the lower central series is key to this determination.

Abstract

We prove that function fields of varieties of dimension at least two over an algebraic closure of a finite field are determined, modulo purely inseparable extensions, by the quotient by the second term in the lower central series of their pro-$\ell$ Galois groups.