Geodesic
Birationally rigid Fano hypersurfaces
Izvestiya. Mathematics , Tome 66 (2002) no. 6, pp. 1243-1269

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We prove that a smooth Fano hypersurface $V=V_M\subset\mathbb P^M$, $M\geqslant 6$, is birationally superrigid. In particular, it cannot be fibred into uniruled varieties by a non-trivial rational map, and every birational map of $V$ onto a minimal Fano variety of the same dimension is a biregular isomorphism. The proof is based on the method of maximal singularities combined with the connectedness principle of Shokurov and Kollar. 
@article{IM2_2002_66_6_a6,
     author = {A. V. Pukhlikov},
     title = {Birationally rigid {Fano} hypersurfaces},
     journal = {Izvestiya. Mathematics },
     pages = {1243--1269},
     publisher = {mathdoc},
     volume = {66},
     number = {6},
     year = {2002},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2002_66_6_a6/}
}
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A. V. Pukhlikov. Birationally rigid Fano hypersurfaces. Izvestiya. Mathematics , Tome 66 (2002) no. 6, pp. 1243-1269. http://geodesic.mathdoc.fr/item/IM2_2002_66_6_a6/