Birational rigidity of Fano hypersurfaces in the framework of Mori theory

This survey reflects the contemporary state of Mori theory and its log version. The main stress is on applications of the theory of log pairs to the birational geometry of varieties of negative Kodaira dimension (as is known, they are close to rational varieties; however, it is also known that many varieties of negative Kodaira dimension are birationally rigid, which is peculiar to a more general class than that of rational varieties), namely, to the Sarkisov program of factorizing birational maps of Mori models that are Mori fibre spaces under the above restrictions. In particular, we present a new proof of the birational rigidity of a non-singular three-dimensional quartic (the Iskovskikh–Manin theorem, which claims that such a quartic is not rational) and of another anticanonical hypersurface in a weighted projective space (from the Corti–Pukhlikov–Reid list). We also present Chel'tsov's results on the birational rigidity of smooth hypersurfaces of degree $N$ in $\mathbb P^N$ for $4\leqslant N\leqslant 8$; the proofs use the Shokurov connectedness theorem.