On the section conjecture over fields of finite type

Assume that the section conjecture holds over number fields. We prove then that it holds for a broad class of curves defined over finitely generated extensions of $\mathbb{Q}$. This class contains every projective, hyperelliptic curve, every hyperbolic, affine curve of genus $\le 2$, and a non-empty open subset of any curve. If we furthermore assume the weak Bombieri-Lang conjecture, we prove that the section conjecture holds for every hyperbolic curve over every finitely generated extension of $\mathbb{Q}$.