The Absolute Anabelian Geometry of Virtual Curves Arising from Sections of Arithmetic Fundamental Groups of Configuration Spaces

Abstract

The objective of this paper is to study the anabelian object referred to as \emph{pointed virtual curves}. Namely, given a family of curves $Y \rightarrow X$ over a field $k$ under suitable conditions, we consider the fundamental-group-theoretic pullback of a Galois section $G_{k} \rightarrow \Pi_{X}$. We show that, in various aspects, such pullbacks exhibit anabelian properties analogous to those of the fundamental group of a curve over $k$.