Algebraic geometry and arithmetic curves /

Algebraic geometry and arithmetic curves /

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Bibliographic Details
Author / Creator:	Liu, Qing, 1963-
Imprint:	Oxford ; New York : Oxford University Press, 2006.
Description:	1 online resource (xv, 577 pages) : illustrations
Language:	English
Series:	Oxford science publications Oxford graduate texts in mathematics ; 6 Oxford science publications. Oxford graduate texts in mathematics ; 6.
Subject:	Curves, Algebraic. Arithmetical algebraic geometry. Arithmetical algebraic geometry. Curves, Algebraic. Electronic books.
Format:	E-Resource Book
URL for this record:	http://pi.lib.uchicago.edu/1001/cat/bib/11217270

Hidden Bibliographic Details
ISBN:	9780191547805 0191547808 9780199202492 0198502842 9780198502845 1281341398 9781281341396
Digital file characteristics:	data file
Notes:	Includes bibliographical references (pages 557-561) and index. English. Print version record.
Summary:	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 th.
Other form:	Print version: Liu, Qing, 1963 July- Algebraic geometry and arithmetic curves. Oxford ; New York : Oxford University Press, 2006 0199202494 9780199202492

Description
Summary:	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.
Physical Description:	1 online resource (xv, 577 pages) : illustrations
Bibliography:	Includes bibliographical references (pages 557-561) and index.
ISBN:	9780191547805 0191547808 9780199202492 0198502842 9780198502845 1281341398 9781281341396