Algebraic Geometry And Arithmetic Curves

Paperback | June 29, 2006

byQing LiuTranslated byErne, Reinie

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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 ofdifferentials, 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 ofreduction 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.

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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 (no...

Qing Liu is Charge de recherche, Centre National de la Recherche Scientifique (CNRS), Laboratoire de Theorie des Nombres et d'Algorithmique Arithmetique, Universite Bordeaux 1.

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Format:PaperbackDimensions:600 pages, 9.21 × 6.14 × 1.28 inPublished:June 29, 2006Publisher:Oxford University PressLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:0199202494

ISBN - 13:9780199202492

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Table of Contents

Introduction1. Some topics in commutative algebra2. General Properties of schemes3. Morphisms and base change4. Some local properties5. Coherent sheaves and Cech cohmology6. Sheaves of differentials7. Divisors and applications to curves8. Birational geometry of surfaces9. Regular surfaces10. Reduction of algebraic curvesBibilographyIndex

Editorial Reviews

`A thorough and far-reaching introduction to algebraic geometry in its scheme-theoretic setting ... The rich bibliography with nearly 100 references enhances the value of this textbook as a great introduction and source for research.'Zentralblatt MATH