  Let U be an open subset of a unirational variety. We prove that there is
rational curve C in U such that the fundamental group of C surjects onto the
fundamental group of U. As a consequence we obtain new proofs of the theorems
of Harbater and Colliot-Th\'el\`ene on Galois covers and torsors over the
p-adic line. We also obtain examples of pencils of curves of genus <14 whose
monodromy group is the full Teichm\"uller group.
