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On the universal curve with unordered marked points in positive characteristic

Ma Luo, Tatsunari Watanabe

TL;DR

This work studies the relative pro-$\ell$ and continuous relative completions of the algebraic fundamental groups of universal curves over the unordered moduli stack in positive characteristic. It proves that for characteristic $p>0$ and odd primes $\ell \neq p$, the natural projection from the universal curve over the unordered moduli stack does not admit a section, hence the corresponding projection on algebraic fundamental groups is non-split. Building on LW25, the authors extend characteristic-zero non-splitting results to positive characteristic by combining specialization arguments, base-change comparisons, and the framework of relative pro-$\ell$ and continuous relative completions. The analysis relies on weight filtrations, monodromy actions, and cohomological descriptions to transfer information between characteristics and to establish non-splitting in the unordered setting. The results deepen understanding of fundamental groups of moduli stacks in positive characteristic and demonstrate how relative and continuous completions illuminate monodromy and splitting phenomena.

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.

On the universal curve with unordered marked points in positive characteristic

TL;DR

This work studies the relative pro- and continuous relative completions of the algebraic fundamental groups of universal curves over the unordered moduli stack in positive characteristic. It proves that for characteristic and odd primes , the natural projection from the universal curve over the unordered moduli stack does not admit a section, hence the corresponding projection on algebraic fundamental groups is non-split. Building on LW25, the authors extend characteristic-zero non-splitting results to positive characteristic by combining specialization arguments, base-change comparisons, and the framework of relative pro- and continuous relative completions. The analysis relies on weight filtrations, monodromy actions, and cohomological descriptions to transfer information between characteristics and to establish non-splitting in the unordered setting. The results deepen understanding of fundamental groups of moduli stacks in positive characteristic and demonstrate how relative and continuous completions illuminate monodromy and splitting phenomena.

Abstract

We study the relative pro- 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.
Paper Structure (22 sections, 15 theorems, 129 equations)

This paper contains 22 sections, 15 theorems, 129 equations.

Table of Contents

  1. Introduction
  2. Moduli of Curves with unordered marked points
  3. Homotopy exact sequences in positive characteristic
  4. Configuration spaces and their fundamental groups
  5. The pro-$\ell$ completion of $\pi_1(F(C, n))$
  6. The $\ell$-adic unipotent completion
  7. Abelianizations of unipotent completions of configuration spaces
  8. Relative Pro-$\ell$ Completion
  9. The relative pro-$\ell$ completion of the mapping class group
  10. Specialization for the universal curve in the ordered case
  11. Specialization for configuration spaces in the ordered case
  12. Homotopy exact sequences in the unordered case
  13. Continuous Relative Completion
  14. Continuous relative completion of $\pi_1({\mathcal{C}}_{g,n/\bar{{\mathbb Q}}}) \to \pi_1({\mathcal{M}}_{g,n/\bar{{\mathbb Q}}})$
  15. The relative completion and relative pro-$\ell$ completion
  16. ...and 7 more sections

Key Result

Theorem 1

Suppose that $\operatorname{char}(k) =p > 0$, $\ell\not=p$ an odd prime, and $g \geq 3$. Then the exact sequence main exact seq in p does not admit any section.

Theorems & Definitions (32)

  • Theorem 1
  • Corollary 2
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • Definition 4.1: Relative pro-$\ell$ completion
  • Proposition 4.2
  • ...and 22 more