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 Q. This class contains every projective, hyperelliptic curve, every hyperbolic, affine curve of genus >= 2, 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 Q.