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Algebraic Geometry and Arithmetic Curves

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ISBN-10: 0198502842

ISBN-13: 9780198502845

Edition: 2002

Authors: Qing Liu, Reinie Erne

List price: $185.00
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This work offers a general introduction to the theory of schemes, followed by applications to arithmetic surfaces and to the theory of reduction of algebraic curves.
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Book details

List price: $185.00
Copyright year: 2002
Publisher: Oxford University Press, Incorporated
Publication date: 7/18/2002
Binding: Hardcover
Pages: 592
Size: 6.14" wide x 9.21" long x 1.44" tall
Weight: 2.156
Language: English

Some topics in commutative algebra
Tensor products
Tensor product of modules
Right-exactness of the tensor product
Tensor product of algebras
Left-exactness: flatness
Local nature of flatness
Faithful flatness
Formal completion
Inverse limits and completions
The Artin-Rees lemma and applications
The case of Noetherian local rings
General properties of schemes
Spectrum of a ring
Zariski topology
Algebraic sets
Ringed topological spaces
Ringed topological spaces
Definition of schemes and examples
Morphisms of schemes
Projective schemes
Noetherian schemes, algebraic varieties
Reduced schemes and integral schemes
Reduced schemes
Irreducible components
Integral schemes
Dimension of schemes
The case of Noetherian schemes
Dimension of algebraic varieties
Morphisms and base change
The technique of base change
Fibered product
Base change
Applications to algebraic varieties
Morphisms of finite type
Algebraic varieties and extension of the base field
Points with values in an extension of the base field
Some global properties of morphisms
Separated morphisms
Proper morphisms
Projective morphisms
Some local properties
Normal schemes
Normal schemes and extensions of regular functions
Regular schemes
Tangent space to a scheme
Regular schemes and the Jacobian criterion
Flat morphisms and smooth morphisms
Flat morphisms
Etale morphisms
Smooth morphisms
Zariski's 'Main Theorem' and applications
Coherent sheaves and Cech cohomology
Coherent sheaves on a scheme
Sheaves of modules
Quasi-coherent sheaves on an affine scheme
Coherent sheaves
Quasi-coherent sheaves on a projective scheme
Cech cohomology
Differential modules and cohomology with values in a sheaf
Cech cohomology on a separated scheme
Higher direct image and flat base change
Cohomology of projective schemes
Direct image theorem
Connectedness principle
Cohomology of the fibers
Sheaves of differentials
Kahler differentials
Modules of relative differential forms
Sheaves of relative differentials (of degree 1)
Differential study of smooth morphisms
Smoothness criteria
Local structure and lifting of sections
Local complete intersection
Regular immersions
Local complete intersections
Duality theory
Canonical sheaf
Grothendieck duality
Divisors and applications to curves
Cartier divisors
Meromorphic functions
Cartier divisors
Inverse image of Cartier divisors
Weil divisors
Cycles of codimension 1
Van der Waerden's purity theorem
Riemann-Roch theorem
Degree of a divisor
Riemann-Roch for projective curves
Algebraic curves
Classification of curves of small genus
Hurwitz formula
Hyperelliptic curves
Group schemes and Picard varieties
Singular curves, structure of Pic[superscript 0](X)
Birational geometry of surfaces
Definition and elementary properties
Universal property of blowing-up
Blowing-ups and birational morphisms
Normalization of curves by blowing-up points
Excellent schemes
Universally catenary schemes and the dimension formula
Cohen-Macaulay rings
Excellent schemes
Fibered surfaces
Properties of the fibers
Valuations and birational classes of fibered surfaces
Regular surfaces
Intersection theory on a regular surface
Local intersection
Intersection on a fibered surface
Intersection with a horizontal divisor, adjunction formula
Intersection and morphisms
Factorization theorem
Projection formula
Birational morphisms and Picard groups
Embedded resolutions
Minimal surfaces
Exceptional divisors and Castelnuovo's criterion
Relatively minimal surfaces
Existence of the minimal regular model
Minimal desingularization and minimal embedded resolution
Applications to contraction; canonical model
Artin's contractability criterion
Determination of the tangent spaces
Canonical models
Weierstrass models and regular models of elliptic curves
Reduction of algebraic curves
Models and reductions
Models of algebraic curves
Reduction map
Reduction of elliptic curves
Reduction of the minimal regular model
Neron models of elliptic curves
Potential semi-stable reduction
Stable reduction of algebraic curves
Stable curves
Stable reduction
Some sufficient conditions for the existence of the stable model
Deligne-Mumford theorem
Simplifications on the base scheme
Proof of Artin-Winters
Examples of computations of the potential stable reduction