Algebraic Geometry and Arithmetic Curves (Oxford Graduate Texts in Mathematics) [Hardcover]

Qing Liu (Author)

Book Description

ISBN-10: 0198502842 | ISBN-13: 978-0198502845 | Publication Date: July 18, 2002 | Edition: 1st

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.

Editorial Reviews

Review

"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

About the Author

Qing Liu is at Charge de recherche, Centre National de la Recherche Scientifique (CNRS), Laboratoire de Theorie des Nombres et d'Algorithmique Arithmetique, Universite Bordeaux 1.

Product Details

• Hardcover: 592 pages
• Publisher: Oxford University Press, USA; 1st edition (July 18, 2002)
• Language: English
• ISBN-10: 0198502842
• ISBN-13: 978-0198502845
• Product Dimensions: 9.3 x 6.1 x 1.4 inches
• Shipping Weight: 2.2 pounds (View shipping rates and policies)
• Average Customer Review: 5.0 out of 5 stars  See all reviews (2 customer reviews)
• Amazon Best Sellers Rank: #2,546,799 in Books (See Top 100 in Books)

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44 of 46 people found the following review helpful:

5.0 out of 5 stars Very good exposition, May 30, 2004

By A Customer

This review is from: Algebraic Geometry and Arithmetic Curves (Oxford Graduate Texts in Mathematics) (Hardcover)

Liu's book has two distinct parts to it. The first 7 chapters combine to give a wonderful exposition of the language of schemes; the other chapters are of a specialised nature and concentrate on arithmetic curves. I will talk about the former. (So when I say "this book", I am only referring to the first 7 chapters)

The book starts off with a chapter on some topics in basic commutative algebra - localisation, flatness and completion. Once this is done, the stage is set to introduce schemes in the next chapter and prove their basic properties. Chapter 3 talks about morphisms of schemes and base change. Chapter 4 continues with a discussion of morphisms and also presents some results about some special types of schemes (normal, regular). It culminates with a proof of Zariski's main theorem. The next chapter takes up sheaf cohomology and is followed up with a chapter on differential calculus on schemes (Kahler differentials, duality theory). Lastly, chapter 7 takes up divisors, proves the Riemann Roch theorem and culminates with some applications to curves.

At a first glance, this would basically look like Hartshorne - the most popular book for an introduction to schemes. However, there are few differences which I will point out. Firstly, Hartshorne emphasizes geometric applications and, as such, uses algebraically closed fields freely. Liu, on the other hand, does not hesistate to give arithmetic applications whenever possible and, therefore, tries to relax the hypotheses on the base field whenever possible. Secondly, Liu is much more readable than Hartshorne which, in its supreme elegance, is a tad dense for a first reading. Unlike Hartshorne, a majority of important results are not presented in the exercises (though many are!). Moreover, unlike Harshorne, this book develops all the necessary commutative algebra along the way (chapter 1,2 of Atiyah-Macdonald should be good enough to read this book). Coming back to the geometry, Hartshorne's chapter 4,5 form an excellent resource for classical geometric applications for theory of schemes. Moreover, chapter 1 presents a very readable and scheme-free account of classical algebraic geometry (pre-Grothendieck) in the language of varieties. Liu's book, however, does not emphasize classical or geometric applications and is not the best place to start if one wishes to learn about varieties.

In the current literature on algebraic geometry, there is a noticeable void. Namely, on one hand, we have Grothendieck's "Elements" (EGA) which present all results about schemes and sheaf cohomology in utmost generality, prove everything with excruciating detail, and are almost unreadable as texts (they're a great references). On the other hand, we have Hartshorne which is basically a beautiful summary of EGA along with geometric applications, but is quite hard to read for an introduction. The book under review is not as concise as Hartshorne's book, presents arithmetic applications and is more readable in a reasonable amount of time than EGA.

In conclusion, this book should be an invaluable resource to anyone who wishes to learn about schemes, especially with arithmetic applications in mind. For those inclined towards geometry, an account of schemes from this book coupled with applications from another book (like Hartshorne) would be a good combination.

1 of 8 people found the following review helpful:

5.0 out of 5 stars Algebraic Geometry and Arithmetic, May 3, 2008

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J. R. BUENDIA "Yoyontzin" (Canada) - See all my reviews
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This review is from: Algebraic Geometry and Arithmetic Curves (Oxford Graduate Texts in Mathematics) (Paperback)

This book together with Matsumura on Commutative Algebra and Hartschone on Algebraic Geometry is an excellent book to learn the subject. I am really enjoying it.