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Rational and nearly rational varieties. (English) Zbl 1060.14073
Cambridge Studies in Advanced Mathematics 92. Cambridge: Cambridge University Press (ISBN 0-521-83207-1/hbk). vi, 235 p. (2004).
This book provides a beautiful and ample introduction to the interesting topic of rational and “nearly rational” varieties and it will be a valuable reference for a wide audience. The authors use both classical and modern methods, in particular they pay careful attention to arithmetic issues. Moreover they give numerous examples and exercises, all of which are accompanied by fully worked-out solutions. The book is divided into seven chapters. Chapter 1 contains some basic examples of rational varieties; cubic surfaces are examined in detail in chapter 2. A general study of rational surfaces is given in chapter 3: classical results are developed within the modern framework of the minimal model program. In chapter 4 examples of higher dimensional smooth nonrational hypersurfaces are constructed, using the method of reduction to prime characteristic. Chapter 5 developes the Noether-Fano method for proving the non-rationality of higher dimensional varieties. Chapter 6 presents the theory of singularities of pairs, with some applications. Chapter 7 contains the solutions of the exercises.

MSC:
14M20 Rational and unirational varieties
14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry
14E08 Rationality questions in algebraic geometry
14J26 Rational and ruled surfaces
14J70 Hypersurfaces and algebraic geometry
14J40 \(n\)-folds (\(n>4\))
14J17 Singularities of surfaces or higher-dimensional varieties
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