Algebraic Geometry and Arithmetic Curves

Liu, Qing

ISBN 10: 0199202494 / ISBN 13: 9780199202492

Published by Oxford : University Press, 2006

About this Item

Original boards, illustrated with numerous equations, 8vo. Oxford Graduate Texts in Mathematics, 6.; Name in pen on title page. Seller Inventory # 342627-ZA27

Bibliographic Details

Title: Algebraic Geometry and Arithmetic Curves

Publisher: Oxford : University Press

Publication Date: 2006

Binding: Soft cover

Condition: Good

About this title

Synopsis:

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.

Review:

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)