Birational rigidity is a striking and
mysterious phenomenon in higher-dimensional algebraic geometry. It
turns out that certain natural families of algebraic varieties (for
example, three-dimensional quartics) belong to the same classification
type as the projective space but have radically different birational
geometric properties. In particular, they admit no non-trivial
birational self-maps and cannot be fibred into rational varieties by a
rational map. The origins of the theory of birational rigidity are in
the work of Max Noether and Fano; however, it was only in 1970 that
Iskovskikh and Manin proved birational superrigidity of quartic
three-folds. This book gives a systematic exposition of, and a
comprehensive introduction to, the theory of birational rigidity,
presenting in a uniform way, ideas, techniques, and results that so
far could only be found in journal papers.
The recent rapid progress in birational geometry and the widening
interaction with the neighboring areas generate the growing interest
to the rigidity-type problems and results. The book brings the reader
to the frontline of current research. It is primarily addressed to
algebraic geometers, both researchers and graduate students, but is
also accessible for a wider audience of mathematicians familiar with
the basics of algebraic geometry.
Readership
Graduate students and research mathematicians interested in
algebraic geometry.