Algebraic Geometry and Arithmetic Curves (Oxford Graduate Texts in Mathematics (0-19-961947-6) Book 6) by Qing Liu (PDF)

Description

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 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 of algebraic curves. The case of elliptic curves is studied in detail. The book concludes with the funadmental 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.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐This book is an excellent companion text for Hartshorne’s algebraic geometry book.By focusing on scheme theory, the book provides a clear and insightful explanation of the core topics in scheme theory. Many concrete examples and counter-examples are discussed throughout the text.Another nice feature of this book is that the exposition is tailored to arithmetic minds, in addition to being a standard textbook on modern algebraic geometry.For example, the definition of algebraic varieties over arbitrary fields allows for a smooth presentation of the pivotal role algebraic geometry plays in algebraic number theory.

⭐Good book overall. Some proofs are not clear because it is done in ad hoc ways. Anyway, this is more readable than Hartshorne’s book and more stuff is going in this book than Shafarevich’s book on scheme. This book only talks about scheme and does not mention old languages at all. Some important topics are not mentioned sufficiently in this book; so it is better to accompany this book with either Hartshorne’s or stack project papers.Also, it does prove most of the commutative algebraic claims in the beginning at least. Eventually the author is forced to just quote the commutative algebraic results, such as CM-rings, excellent rings, etc, but it is not so bad.

⭐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.

⭐tutto perfetto! tranne 2 giorni di ritardo per l’arrivo del libro perchè invece d 5 giorni ci ha impiegato una settimana!

⭐

⭐arithmetic curvesを丁寧に扱っているのは確かにいいです。ただ、基本のスキームとコホモロジーは微妙です。特にコホモロジーについては、チェックコホモロジーしか扱ってないので、エタールコホモロジーを勉強しようとしても、結局ハーツホーン（の３章）は読み直す必要があります。チェック版と、導来関手版は（結局同じものであっても）どっちもやらないといかんです。あと、arithmetic curvesの一番ナイスな例である、楕円曲線のNeron modelについては、SilvermanのNeron modelの章を読むのが、応用を含めていいのかと。ハーツホーンの２章３章読んで、この本でarithmetic curvesを勉強する、というのがいいかも。

⭐

⭐スキーム論の教科書といえばHartshorneがお決まりだったが、数論幾何学を目指す学生さんはLiuの教科書の方を好むようである。眺めてみてその理由がよく分かった。

⭐

Keywords

Free Download Algebraic Geometry and Arithmetic Curves (Oxford Graduate Texts in Mathematics (0-19-961947-6) Book 6) in PDF format
Algebraic Geometry and Arithmetic Curves (Oxford Graduate Texts in Mathematics (0-19-961947-6) Book 6) PDF Free Download
Download Algebraic Geometry and Arithmetic Curves (Oxford Graduate Texts in Mathematics (0-19-961947-6) Book 6) 2006 PDF Free
Algebraic Geometry and Arithmetic Curves (Oxford Graduate Texts in Mathematics (0-19-961947-6) Book 6) 2006 PDF Free Download
Download Algebraic Geometry and Arithmetic Curves (Oxford Graduate Texts in Mathematics (0-19-961947-6) Book 6) PDF
Free Download Ebook Algebraic Geometry and Arithmetic Curves (Oxford Graduate Texts in Mathematics (0-19-961947-6) Book 6)