Birationally rigid varieties with a pencil of double Fano covers. I

The general Fano fibration pi: V --> P-1 the fibre of which is a double Fano hypersurface of index 1 is proved to be birationally superrigid, provided it is sufficiently twisted over the base. In particular, there exist on V no other structures of a rationally convex fibration. The proof is based on the method of maximal singularities.