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Birationally Rigid Varieties
Aleksandr Pukhlikov, University of Liverpool, United Kingdom

Mathematical Surveys and Monographs
2013; 367 pp; hardcover
Volume: 190
ISBN-10: 0-8218-9476-5
ISBN-13: 978-0-8218-9476-7
List Price: US$98
Member Price: US$78.40
Order Code: SURV/190
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See also:

Fundamental Algebraic Geometry: Grothendieck's FGA Explained - Barbara Fantechi, Lothar Gottsche, Luc Illusie, Steven L Kleiman, Nitin Nitsure and Angelo Vistoli

Koszul Cohomology and Algebraic Geometry - Marian Aprodu and Jan Nagel

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.


Graduate students and research mathematicians interested in algebraic geometry.


"The book under review is an introduction to the theory of birational rigidity, and, at the same time, is the first comprehensive account on recent developments of the field."

-- Zentralblatt Math

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