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 (Formula presented.). This class contains every projective, hyperelliptic curve, every hyperbolic, affine curve of genus (Formula presented.), and a basis of open subsets 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 (Formula presented.).