Geodesic
Birationally rigid Fano double hypersurfaces
Sbornik. Mathematics, Tome 191 (2000) no. 6, pp. 883-908

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A general Fano double hypersurface $V$ of index 1 $(\sigma\colon V\to Q_m\subset \mathbb P^{M+1}$ is a double cover branched over a smooth divisor $W=W^*_{2l}\subset\mathbb P^{M+1}$, here $m+l=M+1\geqslant 5)$ is proved to be birationally superrigid; in particular, such a hypersurface admits no non-trivial structures of a fibration into uniruled varieties, and it is non-rational. Its groups of birational and biregular automorphisms coincide. 
@article{SM_2000_191_6_a4,
     author = {A. V. Pukhlikov},
     title = {Birationally rigid {Fano} double hypersurfaces},
     journal = {Sbornik. Mathematics},
     pages = {883--908},
     publisher = {mathdoc},
     volume = {191},
     number = {6},
     year = {2000},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2000_191_6_a4/}
}
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A. V. Pukhlikov. Birationally rigid Fano double hypersurfaces. Sbornik. Mathematics, Tome 191 (2000) no. 6, pp. 883-908. http://geodesic.mathdoc.fr/item/SM_2000_191_6_a4/