Birationally rigid Fano hypersurfaces

Abstract

We prove that a smooth Fano hypersurface , , is birationally superrigid. In particular, it cannot be fibred into uniruled varieties by a non-trivial rational map, and every birational map of 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.