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Algebraic Geometry and Arithmetic Curves (Paperback, New edition)
Loot Price: R2,388
Discovery Miles 23 880
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Algebraic Geometry and Arithmetic Curves (Paperback, New edition)
Series: Oxford Graduate Texts in Mathematics, 6
Expected to ship within 12 - 17 working days
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Total price: R2,408
Discovery Miles: 24 080
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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.
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