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On the Universal Curves with Unordered Marked Points

Tatsunari Watanabe, Ma Luo

TL;DR

The paper proves that the homotopy exact sequence for the universal curve of genus $g$ with $n$ unordered marked points over a characteristic zero field does not split when $g\ge 3$, extending Chen’s topological nonsplitting to the algebraic fundamental group setting. The authors develop and deploy the framework of (continuous) relative completions and their Lie algebras, together with weight filtrations from mixed Hodge theory, to obstruct the existence of algebraic sections. They establish analogous nonsplitting results for the universal hyperelliptic curve, using the hyperelliptic mapping class groups and their symplectic representations. The work also develops detailed symplectic decompositions of the first homology of the relevant relative completions, analyzes $S_n$-equivariant characteristic classes, and provides a robust base-change-compatible toolkit (over ${\mathbb{Q}}$ and ${\mathbb{Q}}_{\ell}$) to study sections and their obstructions. Together, these results illuminate the arithmetic and geometric structure of moduli spaces with unordered marked points and their universal families.

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.

On the Universal Curves with Unordered Marked Points

TL;DR

The paper proves that the homotopy exact sequence for the universal curve of genus with unordered marked points over a characteristic zero field does not split when , extending Chen’s topological nonsplitting to the algebraic fundamental group setting. The authors develop and deploy the framework of (continuous) relative completions and their Lie algebras, together with weight filtrations from mixed Hodge theory, to obstruct the existence of algebraic sections. They establish analogous nonsplitting results for the universal hyperelliptic curve, using the hyperelliptic mapping class groups and their symplectic representations. The work also develops detailed symplectic decompositions of the first homology of the relevant relative completions, analyzes -equivariant characteristic classes, and provides a robust base-change-compatible toolkit (over and ) to study sections and their obstructions. Together, these results illuminate the arithmetic and geometric structure of moduli spaces with unordered marked points and their universal families.

Abstract

Over any field of characteristic , 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.

Paper Structure

This paper contains 23 sections, 22 theorems, 220 equations.

Table of Contents

  1. Introduction
  2. Configuration Spaces of Points on a Curve
  3. Fundamental Groups
  4. Unipotent Completions and Mixed Hodge Structures
  5. Continuous Unipotent Completions
  6. Mapping class groups
  7. Symplectic Representations and Torelli Groups
  8. Moduli space of curves
  9. Homotopy exact sequences
  10. Relative completion
  11. Relative completion and its structure
  12. Properties of relative completion
  13. Relative completion of the mapping class groups
  14. Continuous relative completions over Ql
  15. Generators of Lie algebras of relative completions
  16. ...and 8 more sections

Key Result

Theorem 1

If $g\geq 3$, then the homotopy exact sequence homotopy seq for unordered uni curve does not split.

Theorems & Definitions (38)

  • Theorem 1
  • Theorem 2
  • Theorem 5.1: Hain15
  • Proposition 5.2: Naturality Hain15
  • Proposition 5.3: Right exactness Hain15
  • Proposition 5.4: Base change hain_completion
  • Definition 5.5: Continuous relative completion
  • Proposition 5.6
  • Remark 5.7
  • Example 5.8: Relative completions of $\pi_1^\mathrm{orb}({\mathcal{M}}^\mathrm{an}_{g,n})$ and $\pi_1^\mathrm{orb}({\mathcal{C}}^\mathrm{an}_{g,n})$
  • ...and 28 more