Birational geometry of algebraic varieties with a pencil of Fano complete intersections

We prove birational superrigidity of generic Fano fiber spaces , the fibers of which are Fano complete intersections of index 1 and dimension M in , provided that M ≥ 2k + 1. The proof combines the traditional quadratic techniques of the method of maximal singularities with the linear techniques based on the connectedness principle of Shokurov and Kollár. Certain related results are also considered.