On the Universal Curves with Unordered Marked Points

TL;DR

This paper proves that the homotopy exact sequence of algebraic fundamental groups for universal curves with unordered marked points does not split over any field of characteristic zero, extending topological results to the algebraic setting.

Contribution

It extends Chen's topological nonsplitting result to the profinite algebraic context using relative completions, demonstrating nonexistence of algebraic sections.

Findings

Homotopy exact sequence of algebraic fundamental groups does not split.

Non-splitting holds for universal hyperelliptic curves.

Method extends topological results to algebraic fundamental groups.

Abstract

Over any field of characteristic $0$, we prove that the homotopy exact sequence of algebraic fundamental groups for the universal curve with unordered marked points does not split. The same nonsplitting holds for the universal hyperelliptic curve. Our approach extends Chen's topological result to the profinite setting and relies on the use of relative and continuous relative completions to detect the nonexistence of algebraic sections.