**Mathematical Surveys and Monographs**

Volume: 190;
2013;
367 pp;
Hardcover

MSC: Primary 14;

Print ISBN: 978-0-8218-9476-7

Product Code: SURV/190

List Price: $98.00

Individual Member Price: $78.40

**Electronic ISBN: 978-1-4704-0949-4
Product Code: SURV/190.E**

List Price: $98.00

Individual Member Price: $78.40

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#### Supplemental Materials

# Birationally Rigid Varieties

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*Aleksandr Pukhlikov*

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.

#### Table of Contents

# Table of Contents

## Birationally Rigid Varieties

- Cover Cover11 free
- Title page iii4 free
- Contents v6 free
- Introduction 18 free
- The rationality problem 916 free
- The method of maximal singularities 3744
- Hypertangent divisors 115122
- Rationally connected fibre spaces 153160
- Fano fibre spaces of ℙ¹ 189196
- Del Pezzo fibrations 253260
- Fano direct products 285292
- Double spaces of index two 323330
- Bibliography 359366
- Index 367374 free
- Back Cover Back Cover1377

#### Readership

Graduate students and research mathematicians interested in algebraic geometry.

#### Reviews

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