Higher-dimensional Algebraic Geometry

The classification theory of algebraic varieties is the focus of this book. This very active area of research is still developing, but an amazing quantity of knowledge has accumulated over the past twenty years. The authors goal is to provide an easily accessible introduction to the subject. The book starts with preparatory and standard definitions and results, then moves on to discuss various aspects of the geometry of smooth projective varieties with many rational curves, and finishes in taking the first steps towards Moris minimal model program of classification of algebraic varieties by proving the cone and contraction theorems. The book is well-organized and the author has kept the number of concepts that are used but not proved to a minimum to provide a mostly self-contained introduction.

From the reviews:

"This is an introductory text based on a course the author gave at Harvard University addressing the ... Mori theory. There are essentially two other texts that a student of the subject can consult on this topic. ... The first is a very informal introduction to the theory, while the second is very rigorous, covering the widest range of topics. The book under review lies between these two extremes, balancing rigorous detail with plenty of well-illustrated examples." (Mark Gross, Mathematical Reviews, Issue 2002 g)

"The book is based on a course that the author taught at Harvard University. Concentrating on rational curves was a good idea, since their theory is not treated to the same extent in other introductory books ... . The text is well-written and user-friendly, and contains lots of examples; it is a further good feature that there are exercises at the end of each chapter. ... The book provides a good introduction to higher-dimensional algebraic geometry for graduate students and other interested mathematicians." (Gabor Megyesi, Bulletin of the London Mathematical Society, Issue 35, 2003)

"The book studies the classification theory of algebraic varieties. ... The author’s goal is to provide an easily accessible introduction to the subject. The book begins with preparatory and standard definitions and results, moves on to discuss various aspects of the geometry of smooth projective varieties with many rational curves, and finishes in taking the first steps towards Mori’s minimal model of classification of algebraic varieties by proving the cone and contraction theorems." (L’Enseignement Mathematique, Vol. 48 (1-2), 2002)

"The author’s textbook is based on notes from a class taught at Harvard University. ... He has ... selected suitable parts of the theory and tried to give basic definitions, essential proofs, and important examples with as many details as possible. ... the book provides an excellent source for graduate students ... . The exposition of the material is characterized by a very lucid, refined, and user-friendly style of writing. ... this book fills a gap in the existing textbook literature on algebraic geometry." (Werner Kleinert, Zentralblatt MATH, Vol. 978, 2002)