Algebraic Geometry and Arithmetic Curves

Name: Algebraic Geometry and Arithmetic Curves
Brand: Oxford University Press
Price: 69.09 EUR
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Liu Qing(Autor*in)

Oxford University Press

1. Auflage

Erschienen am 29. Juni 2006

594 Seiten

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978-0-19-154780-5 (ISBN)

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Inhalt

Intro
Contents
1 Some topics in commutative algebra
1.1 Tensor products
1.1.1 Tensor product of modules
1.1.2 Right-exactness of the tensor product
1.1.3 Tensor product of algebras
1.2 Flatness
1.2.1 Left-exactness: flatness
1.2.2 Local nature of flatness
1.2.3 Faithful flatness
1.3 Formal completion
1.3.1 Inverse limits and completions
1.3.2 The Artin-Rees lemma and applications
1.3.3 The case of Noetherian local rings
2 General properties of schemes
2.1 Spectrum of a ring
2.1.1 Zariski topology
2.1.2 Algebraic sets
2.2 Ringed topological spaces
2.2.1 Sheaves
2.2.2 Ringed topological spaces
2.3 Schemes
2.3.1 Definition of schemes and examples
2.3.2 Morphisms of schemes
2.3.3 Projective schemes
2.3.4 Noetherian schemes, algebraic varieties
2.4 Reduced schemes and integral schemes
2.4.1 Reduced schemes
2.4.2 Irreducible components
2.4.3 Integral schemes
2.5 Dimension
2.5.1 Dimension of schemes
2.5.2 The case of Noetherian schemes
2.5.3 Dimension of algebraic varieties
3 Morphisms and base change
3.1 The technique of base change
3.1.1 Fibered product
3.1.2 Base change
3.2 Applications to algebraic varieties
3.2.1 Morphisms of finite type
3.2.2 Algebraic varieties and extension of the base field
3.2.3 Points with values in an extension of the base field
3.2.4 Frobenius
3.3 Some global properties of morphisms
3.3.1 Separated morphisms
3.3.2 Proper morphisms
3.3.3 Projective morphisms
4 Some local properties
4.1 Normal schemes
4.1.1 Normal schemes and extensions of regular functions
4.1.2 Normalization
4.2 Regular schemes
4.2.1 Tangent space to a scheme
4.2.2 Regular schemes and the Jacobian criterion
4.3 Flat morphisms and smooth morphisms
4.3.1 Flat morphisms
4.3.2 Étale morphisms
4.3.3 Smooth morphisms
4.4 Zariski's 'Main Theorem' and applications
5 Coherent sheaves and Cech cohomology
5.1 Coherent sheaves on a scheme
5.1.1 Sheaves of modules
5.1.2 Quasi-coherent sheaves on an affine scheme
5.1.3 Coherent sheaves
5.1.4 Quasi-coherent sheaves on a projective scheme
5.2 Cech cohomology
5.2.1 Differential modules and cohomology with values in a sheaf
5.2.2 Cech cohomology on a separated scheme
5.2.3 Higher direct image and flat base change
5.3 Cohomology of projective schemes
5.3.1 Direct image theorem
5.3.2 Connectedness principle
5.3.3 Cohomology of the fibers
6 Sheaves of differentials
6.1 Kähler differentials
6.1.1 Modules of relative differential forms
6.1.2 Sheaves of relative differentials (of degree 1)
6.2 Differential study of smooth morphisms
6.2.1 Smoothness criteria
6.2.2 Local structure and lifting of sections
6.3 Local complete intersection
6.3.1 Regular immersions
6.3.2 Local complete intersections
6.4 Duality theory
6.4.1 Determinant
6.4.2 Canonical sheaf
6.4.3 Grothendieck duality
7 Divisors and applications to curves
7.1 Cartier divisors
7.1.1 Meromorphic functions
7.1.2 Cartier divisors
7.1.3 Inverse image of Cartier divisors
7.2 Weil divisors
7.2.1 Cycles of codimension 1
7.2.2 Van der Waerden's purity theorem
7.3 Riemann-Roch theorem
7.3.1 Degree of a divisor
7.3.2 Riemann-Roch for projective curves
7.4 Algebraic curves
7.4.1 Classification of curves of small genus
7.4.2 Hurwitz formula
7.4.3 Hyperelliptic curves
7.4.4 Group schemes and Picard varieties
7.5 Singular curves, structure of Pic[sup(0)](X)
8 Birational geometry of surfaces
8.1 Blowing-ups
8.1.1 Definition and elementary properties
8.1.2 Universal property of blowing-up
8.1.3 Blowing-ups and birational morphisms
8.1.4 Normalization of curves by blowing-up points
8.2 Excellent schemes
8.2.1 Universally catenary schemes and the dimension formula
8.2.2 Cohen-Macaulay rings
8.2.3 Excellent schemes
8.3 Fibered surfaces
8.3.1 Properties of the fibers
8.3.2 Valuations and birational classes of fibered surfaces
8.3.3 Contraction
8.3.4 Desingularization
9 Regular surfaces
9.1 Intersection theory on a regular surface
9.1.1 Local intersection
9.1.2 Intersection on a fibered surface
9.1.3 Intersection with a horizontal divisor, adjunction formula
9.2 Intersection and morphisms
9.2.1 Factorization theorem
9.2.2 Projection formula
9.2.3 Birational morphisms and Picard groups
9.2.4 Embedded resolutions
9.3 Minimal surfaces
9.3.1 Exceptional divisors and Castelnuovo's criterion
9.3.2 Relatively minimal surfaces
9.3.3 Existence of the minimal regular model
9.3.4 Minimal desingularization and minimal embedded resolution
9.4 Applications to contraction
canonical model
9.4.1 Artin's contractability criterion
9.4.2 Determination of the tangent spaces
9.4.3 Canonical models
9.4.4 Weierstrass models and regular models of elliptic curves
10 Reduction of algebraic curves
10.1 Models and reductions
10.1.1 Models of algebraic curves
10.1.2 Reduction
10.1.3 Reduction map
10.1.4 Graphs
10.2 Reduction of elliptic curves
10.2.1 Reduction of the minimal regular model
10.2.2 Néron models of elliptic curves
10.2.3 Potential semi-stable reduction
10.3 Stable reduction of algebraic curves
10.3.1 Stable curves
10.3.2 Stable reduction
10.3.3 Some sufficient conditions for the existence of the stable model
10.4 Deligne-Mumford theorem
10.4.1 Simplifications on the base scheme
10.4.2 Proof of Artin-Winters
10.4.3 Examples of computations of the potential stable reduction
Bibliography
Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
Z

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