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.