Algebraic Geometry and Arithmetic Curves

Qing Liu - Collection Oxford Graduate Texts in Mathematics

594 pages, parution le 30/06/2006

Ajouter à une liste

Indisponible

Résumé

Includes essential background methods
Nearly 600 exercises included throughout the text
Lucid, rigorous, coherent and comprehensive exposition
Includes a rich bibliography with nearly 100 references

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.

Readership: Graduate students in algebraic geometry and number theory

Sommaire

Some topics in commutative algebra
General Properties of schemes
Morphisms and base change
Some local properties
Coherent sheaves and Cech cohmology
Sheaves of differentials
Divisors and applications to curves
Birational geometry of surfaces
Regular surfaces
Reduction of algebraic curves

Voir tout

Replier

Caractéristiques techniques

	PAPIER
Éditeur(s)	Oxford University Press
Auteur(s)	Qing Liu
Collection	Oxford Graduate Texts in Mathematics
Parution	30/06/2006
Nb. de pages	594
Format	16 x 23
Couverture	Broché
Poids	890g
Intérieur	Noir et Blanc
EAN13	9780199202492
ISBN13	978-0-19-920249-2