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    Station Eleven

    Emily St. John Mandel 9780385353304

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Oxford Graduate Texts in Mathematics #6: Algebraic Geometry and Arithmetic Curves

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Oxford Graduate Texts in Mathematics #6: Algebraic Geometry and Arithmetic Curves Cover

 

Synopses & Reviews

Publisher Comments:

This new-in-paperback edition provides a general introduction to algebraic and arithmetic geometry, starting with 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.

This book is essentially self-contained, including the necessary material on commutative algebra. The prerequisites are few, and including many examples and approximately 600 exercises, the book is ideal for graduate students.

Table of Contents

1. 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

Product Details

ISBN:
9780199202492
Author:
Liu, Qing
Publisher:
Oxford University Press, USA
Translator:
Erne, Reinie
Author:
null, Qing
Subject:
Geometry - Algebraic
Subject:
Mathematics | Pure Mathematics
Subject:
Mathematics-Algebraic Geometry
Series:
Oxford Graduate Texts in Mathematics
Series Volume:
6
Publication Date:
20060731
Binding:
TRADE PAPER
Grade Level:
Professional and scholarly
Language:
English
Illustrations:
Y
Pages:
600
Dimensions:
6.1 x 9.1 x 1.4 in 1.913 lb

Related Subjects

Science and Mathematics » Biology » Evolution
Science and Mathematics » Mathematics » Geometry » Algebraic Geometry
Science and Mathematics » Mathematics » Topology

Oxford Graduate Texts in Mathematics #6: Algebraic Geometry and Arithmetic Curves New Trade Paper
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Product details 600 pages Oxford University Press, USA - English 9780199202492 Reviews:
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