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

We complete the study of birational geometry of Fano fiber spaces $\pi\colon V\to {\mathbb P}^1$, the fiber of which is a Fano double hypersurface of index

1. For each family of these varieties we either prove birational rigidity or produce explicitly non-trivial structures of Fano fiber spaces. A new linear method of studying movable systems on Fano fiber spaces $V\mathrm{/}{\mathbb{P}}^1$ is developed.