Preface
1. Schemes and sheaves: definitions
1.1. \(\mathrm{Spec}(R)\)
1.2. \(\tilde{M}\)
1.3. Schemes
1.4. Products
1.5. Quasi-coherent sheaves
1.6. The functor of points
1.7. Relativization
1.8. Defining schemes as functors
Appendix: Theory of sheaves
2. Exploring the world of schemes
2.1. Classical varieties as schemes
2.2. The properties: reduced, irreducible and finite type
2.3. Closed subschemes and primary decompositions
2.4. Separated schemes
2.5. \(\mathrm{Proj} R\)
2.6. Proper morphisms
3. Elementary global study of \(\mathrm{Proj} R\)
3.1. Invertible sheaves and twists
3.2. The functor of \(\mathrm{Proj} R\)
3.3. Blow ups
3.4. Quasi-coherent sheaves on \(\mathrm{Proj} R\)
3.5. Ample invertible sheaves
3.6. Invertible sheaves via cocycles, divisors, line bundles
4. Ground fields and base rings
4.1. Kronecker’s big picture
4.2. Galois theory and schemes
4.3. The Frobenius morphism
4.4. Flatness and specialization
4.5. Dimension of fibres of a morphism
4.6. Hensel’s lemma
5. Singular vs. non-singular
5.1. Regularity
5.2. Kähler differential
5.3. Smooth morphisms
5.4. Criteria for smoothness
5.5. Normality
5.6. Zariski’s Main Theorem
5.7. Multiplicities following Weil
6. Group schemes and applications
6.1. Group schemes
6.2. Lang’s theorems over finite fields
7. The cohomology of coherent sheaves
7.1. Basic Čech cohomology
7.2. The case of schemes: Serre’s theorem
7.3. Higher direct images and Leray’s spectral sequence
7.4. Computing cohomology (1): Push \(\mathcal{F}\) into a huge acyclic sheaf
7.5. Computing cohomology (2): Directly via the Čech complex
7.6. Computing cohomology (3): Generate \(\mathcal{F}\) by “known” sheaves
7.7. Computing cohomology (4): Push \(\mathcal{F}\) into a coherent acyclic sheaf
7.8. Serre’s criterion for ampleness
7.9. Functorial properties of ampleness
7.10. The Euler characteristic
7.11. Intersection numbers
7.12. The criterion of Nakai-Moishezon
7.13. Seshadri constants
8. Applications of cohomology
8.1. The Riemann-Roch theorem
Appendix: Residues of differentials on curves
8.2. Comparison of algebraic with analytic cohomology
8.3. De Rham cohomology
8.4. Characteristic p phenomena
8.5. Deformation theory
9. Two deeper results
9.1. Mori’s existence theorem of rational curves
9.2. Belyi’s three point theorem
References
Index