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Algebraic Geometry and Arithmetic Curves (Oxford Graduate Texts in Mathematics)by Qing Liu
Synopses & ReviewsPublisher Comments: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 RiemannRoch 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 blowingups 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 DeligneMumford. The book is essentially selfcontained, 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.
Book News Annotation: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 (booknews.com) Table of Contents1. Some topics in commutative algebra
2. General properties of schemes 3. Morphisms and base change 4. Some local properties 5. Coherent sheaves and Cech cohomology 6. Sheaves of differentials 7. Divisors and applications to curves 8. Birational geometry of surfaces 9. Regular surfaces 10. Reduction of algebraic curves Bibliography Index What Our Readers Are SayingBe the first to add a comment for a chance to win!Product Details
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