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Algebraic Geometry and Arithmetic Curves

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SKU 9780199202492 Categories ,
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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 applicati...

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Description

Product ID:9780199202492
Product Form:Paperback / softback
Country of Manufacture:GB
Series:Oxford Graduate Texts in Mathematics
Title:Algebraic Geometry and Arithmetic Curves
Authors:Author: Qing , Laboratoire de Theorie des Nombres et d'Algorithmique Arithmetique, Universite Bordeaux 1) L, Reinie Erne
Page Count:600
Subjects:Number theory, Number theory, Algebraic geometry, Algebraic topology, Algebraic geometry, Algebraic topology
Description:Select Guide Rating
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.
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.
Imprint Name:Oxford University Press
Publisher Name:Oxford University Press
Country of Publication:GB
Publishing Date:2006-06-29