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Hardcover ISBN:  9780821894767 
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Hardcover ISBN:  9780821894767 
Product Code:  SURV/190 
List Price:  $129.00 
MAA Member Price:  $116.10 
AMS Member Price:  $103.20 
eBook ISBN:  9781470409494 
Product Code:  SURV/190.E 
List Price:  $125.00 
MAA Member Price:  $112.50 
AMS Member Price:  $100.00 
Hardcover ISBN:  9780821894767 
eBook ISBN:  9781470409494 
Product Code:  SURV/190.B 
List Price:  $254.00 $191.50 
MAA Member Price:  $228.60 $172.35 
AMS Member Price:  $203.20 $153.20 

Book DetailsMathematical Surveys and MonographsVolume: 190; 2013; 367 ppMSC: Primary 14
Birational rigidity is a striking and mysterious phenomenon in higherdimensional algebraic geometry. It turns out that certain natural families of algebraic varieties (for example, threedimensional quartics) belong to the same classification type as the projective space but have radically different birational geometric properties. In particular, they admit no nontrivial birational selfmaps 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 threefolds. 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 rigiditytype 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.
ReadershipGraduate students and research mathematicians interested in algebraic geometry.

Table of Contents

Chapters

Introduction

1. The rationality problem

2. The method of maximal singularities

3. Hypertangent divisors

4. Rationally connected fibre spaces

5. Fano fibre spaces of $\mathbb {P}^1$

6. Del Pezzo fibrations

7. Fano direct products

8. Double spaces of index two


Additional Material

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


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Birational rigidity is a striking and mysterious phenomenon in higherdimensional algebraic geometry. It turns out that certain natural families of algebraic varieties (for example, threedimensional quartics) belong to the same classification type as the projective space but have radically different birational geometric properties. In particular, they admit no nontrivial birational selfmaps 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 threefolds. 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 rigiditytype 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.

Chapters

Introduction

1. The rationality problem

2. The method of maximal singularities

3. Hypertangent divisors

4. Rationally connected fibre spaces

5. Fano fibre spaces of $\mathbb {P}^1$

6. Del Pezzo fibrations

7. Fano direct products

8. Double spaces of index two

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