The section conjecture for generic pointed curves
Abstract
Let k/ℚ be finitely generated, let g≥3 and n≥1, and let ℳg,n/k be the moduli stack of smooth proper genus-g curves with n distinct ordered markings. Put K=k(ℳg,n/k). We prove Grothendieck's section conjecture for the generic n-pointed genus-g curve over K. Over both the moduli stack and its function field, the continuous sections of the arithmetic fundamental-group extension of the universal complete curve are, up to geometric conjugacy, exactly the n tautological sections. The same classification holds for every geometrically pro-ℓ quotient, for every prime ℓ. Together with Hain's determination of the rational points, these classifications give examples over fields finitely generated over ℚ with any prescribed positive number of rational points for which the full section conjecture holds. The proof builds on Hain's weighted-completion and nonabelian cohomology methods, with a hyperelliptic calculation providing the genus-three input. An integral normalizer lemma and moving-mark arguments pass from unipotent sections to geometrically pro-ℓ sections. Configuration spaces and Mochizuki's open Hom theorem yield the full profinite classification, and generic descent identifies the section classes over the moduli stack with those over K.
Related
- Derived category of moduli of pointed curves: II
- On the Rank of Normal Functions on Pointed Curves
- A Murphy's law for supports of forgetful maps between moduli spaces of stable pointed curves
- Finiteness of algebraic Weil-Petersson totally geodesic curves in pointed moduli spaces
- Weighted Syzygies of Pointed Curves
BibTeX
@misc{luo2026section,
title = {{The section conjecture for generic pointed curves}},
author = {Ma Luo and Tatsunari Watanabe},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.24986},
eprint = {2610.24986},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.24986},
}
RIS
TY - UNPB
TI - The section conjecture for generic pointed curves
AU - Ma Luo
AU - Tatsunari Watanabe
PY - 2026
DA - 2026/10/18
PB - arXiv
DO - 10.48550/arxiv.2610.24986
UR - https://arxiv.org/abs/2610.24986
AB - Let k/ℚ be finitely generated, let g≥3 and n≥1, and let ℳ_g,n/k be the moduli stack of smooth proper genus-g curves with n distinct ordered markings. Put K=k(ℳ_g,n/k). We prove Grothendieck's section conjecture for the generic n-pointed genus-g curve over K. Over both the moduli stack and its function field, the continuous sections of the arithmetic fundamental-group extension of the universal complete curve are, up to geometric conjugacy, exactly the n tautological sections. The same classification holds for every geometrically pro-ℓ quotient, for every prime ℓ. Together with Hain's determination of the rational points, these classifications give examples over fields finitely generated over ℚ with any prescribed positive number of rational points for which the full section conjecture holds. The proof builds on Hain's weighted-completion and nonabelian cohomology methods, with a hyperelliptic calculation providing the genus-three input. An integral normalizer lemma and moving-mark arguments pass from unipotent sections to geometrically pro-ℓ sections. Configuration spaces and Mochizuki's open Hom theorem yield the full profinite classification, and generic descent identifies the section classes over the moduli stack with those over K.
ER -