The section conjecture for generic pointed curves

Abstract

Let $k/\mathbb Q$ be finitely generated, let $g\geq3$ and $n\geq1$, and let $\mathcal M_{g,n/k}$ be the moduli stack of smooth proper genus-$g$ curves with $n$ distinct ordered markings. Put $K=k(\mathcal M_{g,n/k})$. 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-$\ell$ quotient, for every prime $\ell$. Together with Hain's determination of the rational points, these classifications give examples over fields finitely generated over $\mathbb Q$ 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-$\ell$ 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 $K$.