On the Universal Curves with Unordered Marked Points

2025/11/15 by Tatsunari Watanabe, Watanabe, Tatsunari, Ma Luo +1
Mathematics · Computer Science · #Algebraic Geometry and Number Theory #Homotopy and Cohomology in Algebraic Topology #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2511.12354

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.

Citations

• Remarks on the sections of universal hyperelliptic curves
• On the sections of universal hyperelliptic curves
• The Hodge-de~Rham Theory of Modular Groups
• Relative Weight Filtrations on Completions of Mapping Class Groups
• Moduli stacks of permutation classes of pointed stable curves
• The Hodge de Rham theory of relative Malcev completion
• Infinitesimal presentations of the Torelli groups
• Completions of mapping class groups and the cycle C - C^-

Related

• On cuspidal sections of algebraic fundamental groups
• Remarks on the sections of universal hyperelliptic curves
• On exact sequences of Hodge theoretic fundamental groups
• The Algebraic Proof of the Universality Theorem
• Descent obstruction and fundamental exact sequence