Birationally rigid Fano hypersurfaces with isolated singularities

A. V. Pukhlikov

Sbornik. Mathematics, Tome 193 (2002) no. 3, pp. 445-471

Résumé

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.