Algebraic Geometry and Arithmetic Curves (Paperback)

Qing Liu

ISBN 10: 0199202494 / ISBN 13: 9780199202492

Verlag: Oxford University Press, Oxford, 2006

Beschreibung

Paperback. 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 anda 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 ofRiemann-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 ofalgebraic 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 essentiallyself-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. Shipping may be from our UK warehouse or from our Australian or US warehouses, depending on stock availability. Bestandsnummer des Verkäufers 9780199202492

Bibliografische Details

Titel: Algebraic Geometry and Arithmetic Curves (...

Verlag: Oxford University Press, Oxford

Erscheinungsdatum: 2006

Einband: Paperback

Zustand: new

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Book by Liu Qing

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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)