On the birational section conjecture with strong birationality assumptions

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