On the universal curve with unordered marked points in positive characteristic

TL;DR

This paper investigates the algebraic fundamental groups of universal curves with unordered marked points in positive characteristic, revealing non-splitting properties and extending previous characteristic zero results.

Contribution

It establishes the non-existence of sections for the natural projection of relative completions in positive characteristic, contrasting with characteristic zero cases.

Findings

Proves the non-splitting of the projection in positive characteristic

Compares ordered and unordered fundamental groups using specialization

Extends earlier characteristic zero results to positive characteristic

Abstract

We study the relative pro-$\ell$ and continuous relative completions of the algebraic fundamental groups of universal curves over the moduli stack of curves with unordered marked points in positive characteristic. Using specialization and homotopy exact sequences, we compare the ordered and unordered settings and prove that the natural projection from the relative completion of the universal curve over the unordered moduli stack admits no section in positive characteristic. This yields a non-splitting result for the corresponding projection on algebraic fundamental groups. The present paper is a sequel to our earlier work in characteristic zero.