  A torsor under a k-group scheme G on a variety X over a number field k
imposes a descent obstruction against the existence of rational points on X. We
discuss the finite descent obstruction, that is for all such torsors under
finite k-groups G, in view of a local-global interpolation property for
sections of the fundamental group short exact sequence of X/k. There are
applications to the Brauer-Manin obstruction, to the descent obstruction by
torsors under linear groups, and to the birational version of Grothendieck's
section conjecture over number fields. In particular, we obtain examples of
families of curves over number fields, such that the birational section
conjecture is true in a non-trivial way.
