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OverviewThis 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. Full Product DetailsAuthor: Qing Liu (, Chargé de recherche, Centre National de la Recherche Scientifique (CNRS), Laboratoire de Théorie des Nombres et d'Algorithmique Arithmétique, Université Bordeaux 1) , Erne, ReiniePublisher: Oxford University Press Imprint: Oxford University Press Volume: 6 Dimensions: Width: 15.60cm , Height: 3.30cm , Length: 23.20cm Weight: 0.869kg ISBN: 9780199202492ISBN 10: 0199202494 Pages: 600 Publication Date: 29 June 2006 Audience: Professional and scholarly , Professional & Vocational Format: Paperback Publisher's Status: Active Availability: To order  Stock availability from the supplier is unknown. We will order it for you and ship this item to you once it is received by us. Table of ContentsIntroduction 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 cohmology 6: Sheaves of differentials 7: Divisors and applications to curves 8: Birational geometry of surfaces 9: Regular surfaces 10: Reduction of algebraic curves Bibilography IndexReviews<br> 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<p><br> 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<br> 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 Author InformationTab Content 6Author Website:Countries AvailableAll regions |
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