Algebraic Geometry and Arithmetic Curves

Algebraic Geometry and Arithmetic Curves by Qing Liu

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 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 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.

Review from previous edition Will be useful to graduate students as an introduction to arithmetic algebraic geometry, and to more advanced readers and experts in the field* EMS *
This book is unique in the current literature on algebraic and arithmetic geometry, therefore a highly welcome addition to it, and particularly suitable for readers who want to approach more specialized works in this field with more ease. The exposition is exceptionally lucid, rigorous, coherent and comprehensive. * Zentralblatt MATH *
A thorough and far-reaching introduction to algebraic geometry in its scheme-theoretic setting ... The rich bibliography with nearly 100 references enhances the value of this textbook as a great introduction and source for research. * Zentralblatt MATH *

Yunjun Gao is a professor at the College of Computer Science, Zhejiang University, China. His research interests include database and big data management. He has published more than 90 papers in several leading international journals and conferences including TODS, VLDBJ, TKDE, SIGMOD, VLDB, ICDE, and SIGIR. He is a member of the ACM and the IEEE, and a senior member of the CCF. He was an awardee of the NSFC Excellent Young Scholars Program in 2015, the SIGMOD 2015 Best Paper Nomination, one of the ICDE 2015 Best Papers, the First Prize of the MOE Science and Technology Progress (2016), and the First Prize of the Zhejiang Province Science and Technology (2011).

Qing Liu received his M.S. degree in computer science from Zhejiang University, China, in 2013, and his B.S. degree in software engineering from Zhejiang Normal University, China, in 2010. He completed his Ph.D. degree at the College of Computer Science, Zhejiang University in June 2017 and is currently pursuing postdoctoral research at Hong Kong Baptist University. His research interests include spatial databases and database usability. He has published more than 10 papers in several leading international journals and conferences including VLDBJ, TKDE, VLDB, and ICDE.

SKU	Unavailable
ISBN 13	9780199202492
ISBN 10	0199202494
Title	Algebraic Geometry and Arithmetic Curves
Author	Qing Liu
Series	Oxford Graduate Texts In Mathematics
Condition	Unavailable
Binding Type	Paperback
Publisher	Oxford University Press
Year published	2006-06-29
Number of pages	600
Cover note	Book picture is for illustrative purposes only, actual binding, cover or edition may vary.
Note	Unavailable