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Let $X$ be a curve over a field $k$ finitely generated over $\mathbb{Q}$ and$t$ an indeterminate. We prove that, if $s$ is a section of$\pi_{1}(X)\to\operatorname{Gal}(k)$ such that the base change $s_{k(t)}$ isbirationally liftable, then $s$ comes from geometry. As a consequence we provethat the section conjecture is equivalent to the cuspidalization of allsections over all finitely generated fields.

On the birational section conjecture with strong birationality assumptions