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

TL;DR

This paper investigates the anabelian properties of pointed virtual curves derived from sections of arithmetic fundamental groups, revealing their similarities to classical curve fundamental groups over fields.

Contribution

It introduces the concept of pointed virtual curves and demonstrates their anabelian properties via fundamental-group-theoretic pullbacks, extending classical anabelian geometry.

Findings

Pullbacks exhibit properties similar to fundamental groups of curves over fields

Establishes anabelian characteristics of virtual curves in various aspects

Extends classical anabelian results to virtual curve settings

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 $k$.