Algebraic Geometry and Arithmetic Curves / Edition 1

by Qing Liu

Overview

This book is a general introduction to 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. The book is essentially self-contained, including the necessary material on commutative algebra. The prerequisites are therefore few, and the book should suit a graduate student. It contains many examples and nearly 600 exercises.

Product Details

ISBN-13:	9780198502845
Publisher:	Oxford University Press
Publication date:	07/18/2002
Series:	Oxford Graduate Texts in Mathematics , #6
Pages:	592
Product dimensions:	9.21(w) x 6.14(h) x 1.31(d)

About the Author

Universite Bordeaux 1

Table of Contents

1. Some topics in commutative algebra2. General properties of schemes3. Morphisms and base change4. Some local properties5. Coherent sheaves and Cech cohomology6. Sheaves of differentials7. Divisors and applications to curves8. Birational geometry of surfaces9. Regular surfaces10. Reduction of algebraic curvesBibliographyIndex

Editorial Reviews

A textbook describing the foundation of the geometry of arithmetic surfaces and the theory of stable reduction for undergraduate or graduate students and mathematicians who are not specialists in the field but have a typical undergraduate background in mathematics. Annotation c. Book News, Inc., Portland, OR

"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

From the Publisher