| Hardcover ISBN: | 978-1-4704-1074-2 |
| Product Code: | MMONO/244 |
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| eBook ISBN: | 978-1-4704-1960-8 |
| Product Code: | MMONO/244.E |
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| MAA Member Price: | $139.50 |
| AMS Member Price: | $124.00 |
| Hardcover ISBN: | 978-1-4704-1074-2 |
| eBook: ISBN: | 978-1-4704-1960-8 |
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| AMS Member Price: | $256.00 $194.00 |
| Hardcover ISBN: | 978-1-4704-1074-2 |
| Product Code: | MMONO/244 |
| List Price: | $165.00 |
| MAA Member Price: | $148.50 |
| AMS Member Price: | $132.00 |
| eBook ISBN: | 978-1-4704-1960-8 |
| Product Code: | MMONO/244.E |
| List Price: | $155.00 |
| MAA Member Price: | $139.50 |
| AMS Member Price: | $124.00 |
| Hardcover ISBN: | 978-1-4704-1074-2 |
| eBook ISBN: | 978-1-4704-1960-8 |
| Product Code: | MMONO/244.B |
| List Price: | $320.00 $242.50 |
| MAA Member Price: | $288.00 $218.25 |
| AMS Member Price: | $256.00 $194.00 |
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Book DetailsTranslations of Mathematical MonographsVolume: 244; 2014; 285 ppMSC: Primary 14; 11; 37;
The main goal of this book is to present the so-called birational Arakelov geometry, which can be viewed as an arithmetic analog of the classical birational geometry, i.e., the study of big linear series on algebraic varieties. After explaining classical results about the geometry of numbers, the author starts with Arakelov geometry for arithmetic curves, and continues with Arakelov geometry of arithmetic surfaces and higher-dimensional varieties. The book includes such fundamental results as arithmetic Hilbert–Samuel formula, arithmetic Nakai–Moishezon criterion, arithmetic Bogomolov inequality, the existence of small sections, the continuity of arithmetic volume function, the Lang–Bogomolov conjecture and so on. In addition, the author presents, with full details, the proof of Faltings' Riemann–Roch theorem.
Prerequisites for reading this book are the basic results of algebraic geometry and the language of schemes.
ReadershipGraduate students interested in Diophantine and Arakelov geometry.
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Table of Contents
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Chapters
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Preliminaries
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Geometry of numbers
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Arakelov geometry on arithmetic curves
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Arakelov geometry on arithmetic surfaces
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Arakelov geometry on general arithmetic varieties
-
Arithmetic volume function and its continuity
-
Nakai-Moishezon criterion on an arithmetic variety
-
Arithmetic Bogomolov inequality
-
Lang-Bogomolov conjecture
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Additional Material
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Reviews
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Compared to the earlier books on Arakelov geometry, the current monograph is much more up-to-date, detailed, comprehensive, and self-contained. The exposition stands out of its high degree of clarity, completeness, rigor and topicality, which also makes the volume an excellent textbook on the subject for seasoned graduate students and young researchers in arithmetic algebraic geometry. The rich bibliography of seventy-eight references certainly serves as a useful guide to further reading with regard to the more recent research literature in the field.
Zentralblatt Math -
Many important results are presented for the first time in a book, such as the arithmetic Nakai-Moishezon criterion or the arithmetic Bogomolov inequality. This is a timely monograph that should appeal to researchers in this important area of mathematics.
MAA Reviews
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RequestsReview Copy – for publishers of book reviewsPermission – for use of book, eBook, or Journal contentAccessibility – to request an alternate format of an AMS title
- Book Details
- Table of Contents
- Additional Material
- Reviews
- Requests
The main goal of this book is to present the so-called birational Arakelov geometry, which can be viewed as an arithmetic analog of the classical birational geometry, i.e., the study of big linear series on algebraic varieties. After explaining classical results about the geometry of numbers, the author starts with Arakelov geometry for arithmetic curves, and continues with Arakelov geometry of arithmetic surfaces and higher-dimensional varieties. The book includes such fundamental results as arithmetic Hilbert–Samuel formula, arithmetic Nakai–Moishezon criterion, arithmetic Bogomolov inequality, the existence of small sections, the continuity of arithmetic volume function, the Lang–Bogomolov conjecture and so on. In addition, the author presents, with full details, the proof of Faltings' Riemann–Roch theorem.
Prerequisites for reading this book are the basic results of algebraic geometry and the language of schemes.
Graduate students interested in Diophantine and Arakelov geometry.
-
Chapters
-
Preliminaries
-
Geometry of numbers
-
Arakelov geometry on arithmetic curves
-
Arakelov geometry on arithmetic surfaces
-
Arakelov geometry on general arithmetic varieties
-
Arithmetic volume function and its continuity
-
Nakai-Moishezon criterion on an arithmetic variety
-
Arithmetic Bogomolov inequality
-
Lang-Bogomolov conjecture
-
Compared to the earlier books on Arakelov geometry, the current monograph is much more up-to-date, detailed, comprehensive, and self-contained. The exposition stands out of its high degree of clarity, completeness, rigor and topicality, which also makes the volume an excellent textbook on the subject for seasoned graduate students and young researchers in arithmetic algebraic geometry. The rich bibliography of seventy-eight references certainly serves as a useful guide to further reading with regard to the more recent research literature in the field.
Zentralblatt Math -
Many important results are presented for the first time in a book, such as the arithmetic Nakai-Moishezon criterion or the arithmetic Bogomolov inequality. This is a timely monograph that should appeal to researchers in this important area of mathematics.
MAA Reviews
