Algebraic Geometry
and Arithmetic Curves
Oxford Graduate Texts in Mathematics, 6 (2002),
Oxford University Press,
576 pages, 63 figures.
ISBN 0198502842
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 selfcontained, 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 algebrogeometric part of the book,
it seems fair to say that this is, after A. Grothendieck's voluminous
treatise "Éléments de géométrie algébrique. IIV" (EGA IIV),
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 selfcontained, 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
Contents
 1 Some topics in commutative algebra
 2 General properties of schemes
 3 Morphisms and base change
 4 Some local properties
 5 Coherent sheaves and Cech cohomology
 6 Sheaves of differentials
 7 Divisors and applications to curves
 8 Birational geometry of surfaces
 9 Regular surfaces
 10 Reduction of algebraic curves
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)
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Last update: August 2014