The section conjecture for generic pointed curves

Let $k\mathrm{/}\mathbb{Q}$ be finitely generated, let $g\ge 3$ and $n\ge 1$, and let ${\mathcal{M}}_{g,n\mathrm{/}k}$ be the moduli stack of smooth proper genus-$g$ curves with $n$ distinct ordered markings. Put $K=k({\mathcal{M}}_{g,n\mathrm{/}k})$. 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-$\mathrm{\ell }$ quotient, for every prime $\mathrm{\ell }$. 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-$\mathrm{\ell }$ 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 $K$.