Rational and Nearly Rational Varieties

Modern theory of rational and nearly rational varieties. Many problems with solutions make it ideal for graduate courses.

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 János Kollár, 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.

Introduction; 1. Examples of rational varieties; 2. Cubic surfaces; 3. Rational surfaces; 4. Nonrationality and reduction modulo p; 5. The Noether-Fano method; 6. Singularities of pairs; 7. Solutions to exercises.

Arising from a summer school course, this book develops the modern theory of rational varieties at a level that will particularly suit graduate students. The authors have written a state-of-the-art treatment with numerous exercises and solutions to help students reach the stage where they can begin to tackle contemporary research.