Birationally rigid Fano fibrations

We prove the birational superrigidity of a general Fano fibration $\pi\colon V\to\mathbf P^1$ whose fibre is a Fano hypersurface $W_M\subset\mathbf P^M$ of index 1. If the fibration is sufficiently twisted over the base $\mathbf P^1$, then $V$ has no other structure of a fibration into rationally connected varieties. We also formulate and discuss conjectures on birational rigidity for a large class of Fano varieties and Fano fibrations over a base of arbitrary dimension.