| |
| |
| |
Some topics in commutative algebra | |
| |
| |
| |
Tensor products | |
| |
| |
| |
Tensor product of modules | |
| |
| |
| |
Right-exactness of the tensor product | |
| |
| |
| |
Tensor product of algebras | |
| |
| |
| |
Flatness | |
| |
| |
| |
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 | |
| |
| |
| |
Sheaves | |
| |
| |
| |
Ringed topological spaces | |
| |
| |
| |
Schemes | |
| |
| |
| |
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 | |
| |
| |
| |
Dyimension 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 | |
| |
| |
| |
Frobenius | |
| |
| |
| |
Some global properties of morphisms | |
| |
| |
| |
Separated morphisms | |
| |
| |
| |
Proper morphisms | |
| |
| |
| |
Projective morphisms | |
| |
| |
| |
Some local properties | |
| |
| |
| |
Normal schemes | |
| |
| |
| |
Normal schemes and extensions of regular functions | |
| |
| |
| |
Normalization | |
| |
| |
| |
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 | |
| |
| |
| |
Determinant | |
| |
| |
| |
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[supercript 0] (X) | |
| |
| |
| |
Birational geometry of surfaces | |
| |
| |
| |
Blowing-ups | |
| |
| |
| |
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 | |
| |
| |
| |
Contraction | |
| |
| |
| |
Desingularization | |
| |
| |
| |
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 | |
| |
| |
| |
Reduction map | |
| |
| |
| |
Graphs | |
| |
| |
| |
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 | |
| |
| |
Bibliography | |
| |
| |
Index | |