Abstract
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 and index over a rationally connected base of dimension at most . 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.
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