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Oxford Graduate Texts in Mathematics Ser.: Algebraic Geometry and Arithmetic Curves by Qing Liu (2006, Perfect)
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About this product
Product Identifiers
PublisherOxford University Press, Incorporated
ISBN-100199202494
ISBN-139780199202492
eBay Product ID (ePID)52633552
Product Key Features
Number of Pages600 Pages
Publication NameAlgebraic Geometry and Arithmetic Curves
LanguageEnglish
Publication Year2006
SubjectGeometry / Algebraic
TypeTextbook
AuthorQing Liu
Subject AreaMathematics
SeriesOxford Graduate Texts in Mathematics Ser.
FormatPerfect
Dimensions
Item Height1.3 in
Item Weight30.7 Oz
Item Length9.1 in
Item Width6.1 in
Additional Product Features
Intended AudienceScholarly & Professional
Dewey Edition21
Reviews"Although other books do offer a fast passage to modern number theory, ... only Liu provides a systematic development of algebraic geometry aimed at arithmetic."--Choice, '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.'EMS'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.'Zentralblatt MATH'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, '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.'EMS, '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, '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.'Zentralblatt MATH
Series Volume Number6
IllustratedYes
Dewey Decimal516.352
Table Of ContentIntroduction1. 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
SynopsisThis 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., 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 desingularization (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.
LC Classification NumberQA564
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