Algebraic geometry and arithmetic curves

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Algebraic geometry and arithmetic curves / Qing Liu ; translated by Reinie Erné

資料種別:

図書

出版情報:

Oxford : Oxford University Press, 2006

形態:

xv, 577 p. ; 24 cm

シリーズ名:

Oxford graduate texts in mathematics ; 6 <BA31148509>

著者名:

ISBN:

9780199202492 [0199202494] (: pbk)

書誌ID:

BA77903549

子書誌情報

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他の版・巻

Algebraic geometry and arithmetic curves
Qing Liu ; translated by Reinie Erné , Oxford University Press , 2002

書誌詳細

注記:

Translated from the French
First published 2002
Bibliography: p. [557]-561
Includes index

主題:

Curves; Geometry, Algebraic

言語:

英語

目次情報:

Some topics in commutative algebra / 1：

Tensor products / 1.1：

Tensor product of modules / 1.1.1：

Right-exactness of the tensor product / 1.1.2：

Tensor product of algebras / 1.1.3：

Flatness / 1.2：

Left-exactness: flatness / 1.2.1：

Local nature of flatness / 1.2.2：

Faithful flatness / 1.2.3：

Formal completion / 1.3：

Inverse limits and completions / 1.3.1：

The Artin-Rees lemma and applications / 1.3.2：

The case of Noetherian local rings / 1.3.3：

General properties of schemes / 2：

Spectrum of a ring / 2.1：

Zariski topology / 2.1.1：

Algebraic sets / 2.1.2：

Ringed topological spaces / 2.2：

Sheaves / 2.2.1：

Schemes / 2.2.2：

Definition of schemes and examples / 2.3.1：

Morphisms of schemes / 2.3.2：

Projective schemes / 2.3.3：

Noetherian schemes, algebraic varieties / 2.3.4：

Reduced schemes and integral schemes / 2.4：

Reduced schemes / 2.4.1：

Irreducible components / 2.4.2：

Integral schemes / 2.4.3：

Dimension / 2.5：

Dyimension of schemes / 2.5.1：

The case of Noetherian schemes / 2.5.2：

Dimension of algebraic varieties / 2.5.3：

Morphisms and base change / 3：

The technique of base change / 3.1：

Fibered product / 3.1.1：

Base change / 3.1.2：

Applications to algebraic varieties / 3.2：

Morphisms of finite type / 3.2.1：

Algebraic varieties and extension of the base field / 3.2.2：

Points with values in an extension of the base field / 3.2.3：

Frobenius / 3.2.4：

Some global properties of morphisms / 3.3：

Separated morphisms / 3.3.1：

Proper morphisms / 3.3.2：

Projective morphisms / 3.3.3：

Some local properties / 4：

Normal schemes / 4.1：

Normal schemes and extensions of regular functions / 4.1.1：

Normalization / 4.1.2：

Regular schemes / 4.2：

Tangent space to a scheme / 4.2.1：

Regular schemes and the Jacobian criterion / 4.2.2：

Flat morphisms and smooth morphisms / 4.3：

Flat morphisms / 4.3.1：

Etale morphisms / 4.3.2：

Smooth morphisms / 4.3.3：

Zariski's 'Main Theorem' and applications / 4.4：

Coherent sheaves and Cech cohomology / 5：

Coherent sheaves on a scheme / 5.1：

Sheaves of modules / 5.1.1：

Quasi-coherent sheaves on an affine scheme / 5.1.2：

Coherent sheaves / 5.1.3：

Quasi-coherent sheaves on a projective scheme / 5.1.4：

Cech cohomology / 5.2：

Differential modules and cohomology with values in a sheaf / 5.2.1：

Cech cohomology on a separated scheme / 5.2.2：

Higher direct image and flat base change / 5.2.3：

Cohomology of projective schemes / 5.3：

Direct image theorem / 5.3.1：

Connectedness principle / 5.3.2：

Cohomology of the fibers / 5.3.3：

Sheaves of differentials / 6：

Kahler differentials / 6.1：

Modules of relative differential forms / 6.1.1：

Sheaves of relative differentials (of degree 1) / 6.1.2：

Differential study of smooth morphisms / 6.2：

Smoothness criteria / 6.2.1：

Local structure and lifting of sections / 6.2.2：

Local complete intersection / 6.3：

Regular immersions / 6.3.1：

Local complete intersections / 6.3.2：

Duality theory / 6.4：

Determinant / 6.4.1：

Canonical sheaf / 6.4.2：

Grothendieck duality / 6.4.3：

Divisors and applications to curves / 7：

Cartier divisors / 7.1：

Meromorphic functions / 7.1.1：

Inverse image of Cartier divisors / 7.1.2：

Weil divisors / 7.2：

Cycles of codimension 1 / 7.2.1：

Van der Waerden's purity theorem / 7.2.2：

Riemann-Roch theorem / 7.3：

Degree of a divisor / 7.3.1：

Riemann-Roch for projective curves / 7.3.2：

Algebraic curves / 7.4：

Classification of curves of small genus / 7.4.1：

Hurwitz formula / 7.4.2：

Hyperelliptic curves / 7.4.3：

Group schemes and Picard varieties / 7.4.4：

Singular curves, structure of Pic[supercript 0] (X) / 7.5：

Birational geometry of surfaces / 8：

Blowing-ups / 8.1：

Definition and elementary properties / 8.1.1：

Universal property of blowing-up / 8.1.2：

Blowing-ups and birational morphisms / 8.1.3：

Normalization of curves by blowing-up points / 8.1.4：

Excellent schemes / 8.2：

Universally catenary schemes and the dimension formula / 8.2.1：

Cohen-Macaulay rings / 8.2.2：

Fibered surfaces / 8.2.3：

Properties of the fibers / 8.3.1：

Valuations and birational classes of fibered surfaces / 8.3.2：

Contraction / 8.3.3：

Desingularization / 8.3.4：

Regular surfaces / 9：

Intersection theory on a regular surface / 9.1：

Local intersection / 9.1.1：

Intersection on a fibered surface / 9.1.2：

Intersection with a horizontal divisor, adjunction formula / 9.1.3：

Intersection and morphisms / 9.2：

Factorization theorem / 9.2.1：

Projection formula / 9.2.2：

Birational morphisms and Picard groups / 9.2.3：

Embedded resolutions / 9.2.4：

Minimal surfaces / 9.3：

Exceptional divisors and Castelnuovo's criterion / 9.3.1：

Relatively minimal surfaces / 9.3.2：

Existence of the minimal regular model / 9.3.3：

Minimal desingularization and minimal embedded resolution / 9.3.4：

Applications to contraction; canonical model / 9.4：

Artin's contractability criterion / 9.4.1：

Determination of the tangent spaces / 9.4.2：

Canonical models / 9.4.3：

Weierstrass models and regular models of elliptic curves / 9.4.4：

Reduction of algebraic curves / 10：

Models and reductions / 10.1：

Models of algebraic curves / 10.1.1：

Reduction / 10.1.2：

Reduction map / 10.1.3：

Graphs / 10.1.4：

Reduction of elliptic curves / 10.2：

Reduction of the minimal regular model / 10.2.1：

Neron models of elliptic curves / 10.2.2：

Potential semi-stable reduction / 10.2.3：

Stable reduction of algebraic curves / 10.3：

Stable curves / 10.3.1：

Stable reduction / 10.3.2：

Some sufficient conditions for the existence of the stable model / 10.3.3：

Deligne-Mumford theorem / 10.4：

Simplifications on the base scheme / 10.4.1：

Proof of Artin-Winters / 10.4.2：

Examples of computations of the potential stable reduction / 10.4.3：

Bibliography

Index

Some topics in commutative algebra / 1：

Tensor products / 1.1：

Tensor product of modules / 1.1.1：

Right-exactness of the tensor product / 1.1.2：

Tensor product of algebras / 1.1.3：

Flatness / 1.2：

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