Birational geometry of algebraic varieties fibred into Fano double spaces

We develop a quadratic technique for proving the birational rigidity of Fano-Mori fibre spaces over a higher-dimensional base. As an application, we prove the birational rigidity of generic fibrations into Fano double spaces of dimension M >= 4 and index 1 over a rationally connected base of dimension at most ( M - 2)(M - 1)/2. We obtain a near-optimal estimate for the codimension of the set of hypersurfaces of a given degree in projective space that have positive-dimensional singular sets.