Algebraic Geometry and Arithmetic Curves

Oxford Graduate Texts in Mathematics, 6 (2002),
Oxford University Press,
576 pages, 63 figures.
ISBN 0-19-850284-2

Paperback new edition (2006)
Oxford University Press,
ISBN 0-19-920249-4

China edition (2013)
World Publishing Corporation, Beijing
ISBN 978-7-5100-4413-7

This book provides a general introduction to the theory of schemes, followed by applications to arithmetic surfaces and to the theory of reduction of algebraic curves.

The book is essentially self-contained, including the necessary material on commutative algebra. The prerequisites are therefore few, and the book should suit a graduate student. It contains many examples and nearly six hundred exercises.

Prof. Dr. Werner Kleinert (Berlin) wrote in Zentralblatt Math.:

As to the first, purely algebro-geometric part of the book, it seems fair to say that this is, after A. Grothendieck's voluminous treatise "Éléments de Géométrie Algébrique. I--IV" (EGA I--IV), the most comprehensive and detailed elaboration of the theory of algebraic schemes available in (text-)book form, whereas the second, merely arithmetic part provides the very first systematic and coherent introduction to the advanced theory of arithmetic curves and surfaces at all. Moreover, the entire text is arranged in such exhaustive a way that the book is essentially self-contained, keeping the prerequisites at a minimum, and perfectly suitable for seasoned graduate students. Another feature of this highly valuable book on algebraic and arithmetic geometry is provided by the vast amount of illustrating, theoretically important examples as well as by the approximately six hundred included exercises.
As for the study of algebraic varieties, there are many other excellent (specific) textbooks that can be consulted. As stated before, this book is unique in the current literature on algebraic and arithmetic geometry, therefore a highly welcome addition to it, and particularly suitable for readers who want to approach more specialized works in this field with more ease. The exposition is exceptionally lucid, rigorous, coherent and comprehensive, in addition to all the other mentioned advantages of the book.

Preface/ Preface to the paperback edition  


Complete Table of Contents

Index of terminology

Errata for the hardcover edition (2002)

Errata for the paperback edition (2006) (version of November 2008)

second errata/addenda for the paperback edition (2006) (version of Aug. 2010)

Last update:   August 2014