On the birational section conjecture with strong birationality assumptions

Abstract

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 ${\pi }_1(X)\to \mathrm{Gal}(k)$ such that the base change $s_{k(t)}$ is birationally liftable, then $s$ 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.

Final version

References in corpus (1)

• An arithmetic valuative criterion for proper maps of tame algebraic stacks

Cited by in corpus (1)

• The genericity theorem for the essential dimension of tame stacks