Geodesic
Birational geometry of algebraic varieties fibred into Fano double spaces
Izvestiya. Mathematics , Tome 81 (2017) no. 3, pp. 618-644

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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.
Keywords: Fano variety, maximal singularity, linear system.
Mots-clés : Fano–Mori fibre space, birational map 
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     author = {A. V. Pukhlikov},
     title = {Birational geometry of algebraic varieties fibred into {Fano} double spaces},
     journal = {Izvestiya. Mathematics },
     pages = {618--644},
     publisher = {mathdoc},
     volume = {81},
     number = {3},
     year = {2017},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2017_81_3_a6/}
}
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A. V. Pukhlikov. Birational geometry of algebraic varieties fibred into Fano double spaces. Izvestiya. Mathematics , Tome 81 (2017) no. 3, pp. 618-644. http://geodesic.mathdoc.fr/item/IM2_2017_81_3_a6/