Rational and Nearly Rational Varieties

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Cambridge University Press, 2004/04/22 - 235 ページ
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The most basic algebraic varieties are the projective spaces, and rational varieties are their closest relatives. In many applications where algebraic varieties appear in mathematics and the sciences, we see rational ones emerging as the most interesting examples. The authors have given an elementary treatment of rationality questions using a mix of classical and modern methods. Arising from a summer school course taught by Jnos Kollr, this book develops the modern theory of rational and nearly rational varieties at a level that will particularly suit graduate students. There are numerous examples and exercises, all of which are accompanied by fully worked out solutions, that will make this book ideal as the basis of a graduate course. It will act as a valuable reference for researchers whilst helping graduate students to reach the point where they can begin to tackle contemporary research problems.
 

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目次

Introduction
1
Quadrics over finite fields
14
Further examples of rational varieties
29
The proofs of the theorems of Segre and Manin
42
Rational surfaces
60
Field of definition of a subvariety
76
Nonratiimulity via reduction modulo p
93
The NoetherFano method for proving nonrationality
122
Singularities of pairs
149
Solutions to exercises
186
References
228
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