Algebraic Geometry and Arithmetic Curves
Series: Oxford Graduate Texts in Mathematics; 6;
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Product details:
- Edition number New ed
- Publisher OUP Oxford
- Date of Publication 29 June 2006
- ISBN 9780199202492
- Binding Paperback
- No. of pages600 pages
- Size 232x156x32 mm
- Weight 869 g
- Language English 0
Categories
Short description:
This new-in-paperback edition provides an introduction to algebraic and arithmetic geometry, starting with the theory of schemes, followed by applications to arithmetic surfaces and to the theory of reduction of algebraic curves. Clear explanations of both theory and applications, and almost 600 exercises are included in the text.
MoreLong description:
This new-in-paperback edition provides a general introduction to algebraic and arithmetic geometry, starting with the theory of schemes, followed by applications to arithmetic surfaces and to the theory of reduction of algebraic curves.
The first part introduces basic objects such as schemes, morphisms, base change, local properties (normality, regularity, Zariski's Main Theorem). This is followed by the more global aspect: coherent sheaves and a finiteness theorem for their cohomology groups. Then follows a chapter on sheaves of differentials, dualizing sheaves, and Grothendieck's duality theory. The first part ends with the theorem of Riemann-Roch and its application to the study of smooth projective curves over a field. Singular curves are treated through a detailed study of the Picard group.
The second part starts with blowing-ups and desingularisation (embedded or not) of fibered surfaces over a Dedekind ring that leads on to intersection theory on arithmetic surfaces. Castelnuovo's criterion is proved and also the existence of the minimal regular model. This leads to the study of reduction of algebraic curves. The case of elliptic curves is studied in detail. The book concludes with the fundamental theorem of stable reduction of Deligne-Mumford.
This book is essentially self-contained, including the necessary material on commutative algebra. The prerequisites are few, and including many examples and approximately 600 exercises, the book is ideal for graduate students.
Review from previous edition Will be useful to graduate students as an introduction to arithmetic algebraic geometry, and to more advanced readers and experts in the field.
Table of Contents:
Introduction
Some topics in commutative algebra
General Properties of schemes
Morphisms and base change
Some local properties
Coherent sheaves and Cech cohmology
Sheaves of differentials
Divisors and applications to curves
Birational geometry of surfaces
Regular surfaces
Reduction of algebraic curves
Bibilography
Index
