Preface Part 1. Grothendieck topologies~ fibered categories and descent theory Introduction Chapter 1. Preliminary notions 1.1. Algebraic geometry 1.2. Category theory Chapter 2. Contravariant functors 2.1. Representable functors and the Yoneda Lemma 2.2. Group objects 2.3. Sheaves in Grothendieck topologies Chapter 3. Fibered categories 3.1. Fibered categories 3.2. Examples of fibered categories 3.3. Categories fibered in groupoids 3.4. Functors and categories fibered in sets 3.5. Equivalences of fibered categories 3.6. Objects as fibered categories and the 2-Yoneda Lemma 3.7. The functors of arrows of a fibered category 3.8. Equivariant objects in fibered categories Chapter 4. Stacks 4.1. Descent of objects of fibered categories 4.2. Descent theory for quasi-coherent sheaves 4.3. Descent for morphisms of schemes 4.4. Descent along torsors Part 2. Construction of Hilbert and Quot schemes Chapter 5. Construction of Hilbert and Quot schemes Introduction 5.1. The Hilbert and Quot functors 5.2. Castelnuovo-Mumford regularity 5.3. Semi-continuity and base-change 5.4, Generic flatness and flattening stratification 5.5. Construction of Quot schemes 5.6. Some variants and applications Part 3. Local properties and Hilbert schemes of points Introduction Chapter 6. Elementary Deformation Theory 6.1. Infinitesimal study of schemes 6.2. Pro-representable functors 6.3. Non-pro-representable functors 6.4, Examples of tangent-obstruction theories 6.5. More tangent-obstruction theories Chapter 7. Hilbert Schemes of Points Introduction 7.1. The symmetric power and the Hilbert-Chow morphism 7.2. Irreducibility and nonsingularity 7.3. Examples of Hilbert schemes 7.4. A stratification of the Hilbert schemes 7.5. The Betti numbers of the Hilbert schemes of points 7.6. The Heisenberg algebra Part 4. Grothendieck's existence theorem in formal geometry with a letter of Jean-Pierre Serre Chapter 8. Grothendieck's existence theorem in formal geometry Introduction 8.1. Locally noetherian formal schemes 8.2. The comparison theorem 8.3. Cohomological flatness 8.4. The existence theorem 8.5. Applications to lifting problems 8.6. Serre's examples 8.7. A letter of Serre Part 5. The Picard scheme Chapter 9. The Picard scheme 9.1. Introduction 9.2. The several Picard functors 9.3. Relative effective divisors 9.4. The Picard scheme 9.5. The connected component of the identity 9.6. The torsion component of the identity Appendix A. Answers to all the exercises Appendix B. Basic intersection theory Bibliography Index