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 $M\geqslant 4$ 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.