Geodesic
Birationally rigid Fano hypersurfaces with isolated singularities
Sbornik. Mathematics, Tome 193 (2002) no. 3, pp. 445-471 
A. V. Pukhlikov. Birationally rigid Fano hypersurfaces with isolated singularities. Sbornik. Mathematics, Tome 193 (2002) no. 3, pp. 445-471. http://geodesic.mathdoc.fr/item/SM_2002_193_3_a8/
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     title = {Birationally rigid {Fano} hypersurfaces with isolated singularities},
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     url = {http://geodesic.mathdoc.fr/item/SM_2002_193_3_a8/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

It is proved that a general Fano hypersurface $V=V_M\subset{\mathbb P}^M$ of index 1 with isolated singularities in general position is birationally rigid. Hence it cannot be fibred into uniruled varieties of smaller dimension by a rational map, and each ${\mathbb Q}$-Fano variety $V'$ with Picard number 1 birationally equivalent to $V$ is in fact isomorphic to $V$. In particular, $V$ is non-rational. The group of birational self-maps of $V$ is either {1} or ${\mathbb Z}/2{\mathbb Z}$, depending on whether $V$ has a terminal point of the maximum possible multiplicity $M- 2$. The proof is based on a combination of the method of maximal singularities and the techniques of hypertangent systems with Shokurov's connectedness principle.

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