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