Oxford Graduate Texts in Mathematics (0-19-961947-6) – serie
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2 produkter
2 produkter
Del 6 - Oxford Graduate Texts in Mathematics (0-19-961947-6)
Algebraic Geometry and Arithmetic Curves
Inbunden, Engelska, 2002
1 973 kr
Skickas inom 5-8 vardagar
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.
Del 9 - Oxford Graduate Texts in Mathematics (0-19-961947-6)
Topology: A Geometric Approach
Inbunden, Engelska, 2003
1 136 kr
Skickas inom 5-8 vardagar
The book gives an introduction to topology at the advanced undergraduate to beginning graduate level, with an emphasis on its geometric aspects. Part I contains three chapters on basic point set topology, classification of surfaces via handle decompositions, and the fundamental group appropriate for a one semester or two quarter course. Carefully developed exercise sets support high student involvement. Besides exercises embedded within a chapter, there are extensive supplementary exercises to extend the material. Surfaces occur as key examples in treatments of the fundamental group, covering spaces, CW complexes, and homology in the last four chapters. Each chapter of Part I ends with a substantial project. Part II is written in a problem based format. These problems contain appropriate hints and background material to enable the student to work through the basic theory of covering spaces, CW complexes, and homology with the instructor's guidance. Low dimensional cases provide motivation and examples for the general development, with an emphasis on treating geometric ideas first encountered in Part I such as orientation. Part II allows the book to be used for a year long course at the first year graduate level.The book's collection of over 750 exercises range from simple checks of omitted details in arguments, to reinforce the material and increase student involvement, to the development of substantial theorems that have been broken into many steps. The style encourages an active student role. Solutions to selected exercises are included as an appendix. Solutions to all exercises are available to the instructor in electronic form.This text forms the latest in the Oxford Graduate Texts in Mathematics series which publishes textbooks suitable for graduate students in mathematics and its applications. The level of books may range from some suitable for advanced undergraduate courses at one end, to others of interest to research workers. The emphasis is on texts of high mathematical quality in active areas, particularly areas that are not well represented in the literature at present.