  Let X be a smooth projective curve of genus >1 over a field K which is
finitely generated over the rationals. The section conjecture in Grothendieck's
anabelian geometry says that the sections of the canonical projection from the
arithmetic fundamental group of X onto the absolute Galois group of K are (up
to conjugation) in one-to-one correspondence with K-rational points of X. The
birational variant conjectures a similar correspondence where the fundamental
group is replaced by the absolute Galois group of the function field K(X).
  The present paper proves the birational section conjecture for all X when K
is replaced e.g. by the field of p-adic numbers. It disproves both conjectures
for the field of real or p-adic algebraic numbers. And it gives a purely group
theoretic characterization of the sections induced by K-rational points of X in
the birational setting over almost arbitrary fields.
  As a biproduct we obtain Galois theoretic criteria for radical solvability of
polynomial equations in more than one variable, and for a field to be PAC, to
be large, or to be Hilbertian.
