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

The study of the birational geometry of Fano fibre spaces with fibre that is a Fano double hypersurface of index 1 is completed. For each family of these varieties either the birational rigidity is proved or non-trivial structures of Fano fibre spaces are explicitly produced. A new linear method of the study of movable systems on Fano fibre spaces is developed.