A First Course in Algebraic Geometry and Algebraic Varieties

Contents
Preface
About the Author
Acknowledgments
1. Basics on Commutative Algebra
1.1 Ideals and Operations on Ideals
1.2 UFDs and PIDs
1.3 Polynomial Rings
1.3.1 Polynomials in D[x], where D a UFD
1.3.2 The case D = K a field
1.3.3 Resultant of two polynomials in D[x]
1.3.4 Resultant in D[x1, . . . ,xn] and elimination
1.4 Noetherian Rings and the Hilbert Basis Theorem
1.5 R-Modules, R-Algebras and Finiteness Conditions
1.6 Integrality
1.7 Zariski’s Lemma
1.8 Transcendence Degree
1.9 Tensor Products of R-Modules and of R-Algebras
1.9.1 Restriction and extension of scalars
1.9.2 Tensor product of algebras
1.10 Graded Rings and Modules, Homogeneous Ideals
1.10.1 Homogeneous polynomials
1.10.2 Graded modules and graded morphisms
1.11 Localization
1.11.1 Local rings and localization
1.12 Krull-Dimension of a Ring
Exercises
2. Algebraic Affine Sets
2.1 Algebraic Affine Sets and Ideals
2.2 Hilbert “Nullstellensatz”
2.3 Some Consequences of Hilbert “Nullstellensatz” and of Elimination Theory
2.3.1 Study’s principle
2.3.2 Intersections of affine plane curves
Exercises
3. Algebraic Projective Sets
3.1 Algebraic Projective Sets
3.2 Homogeneous “Hilbert Nullstellensatz”
3.3 Fundamental Examples and Remarks
3.3.1 Points
3.3.2 Coordinate linear subspaces
3.3.3 Hyperplanes and the dual projective space
3.3.4 Fundamental affine open sets (or affine charts) of Pn
3.3.5 Projective closure of affine sets
3.3.6 Projective subspaces and their ideals
3.3.7 Projective and affine subspaces
3.3.8 Homographies, projectivities and affinities
3.3.9 Projective cones
3.3.10 Projective hypersurfaces and projective closure of affine hypersurfaces
3.3.11 Proper closed subsets of P2
3.3.12 Affine and projective twisted cubics
Exercises
4. Topological Properties and Algebraic Varieties
4.1 Irreducible Topological Spaces
4.1.1 Coordinate rings, ideals and irreducibility
4.1.2 Algebraic varieties
4.2 Noetherian Spaces: Irreducible Components
4.3 Combinatorial Dimension
Exercises
5. Regular and Rational Functions on Algebraic Varieties
5.1 Basics on Sheaves
5.2 Regular Functions
5.3 Rational Functions
5.3.1 Consequences of the fundamental theorem on regular and rational functions
5.3.2 Examples
Exercises
6. Morphisms of Algebraic Varieties
6.1 Morphisms
6.2 Morphisms with (Quasi) Affine Target
6.3 Morphisms with (Quasi) Projective Target
6.4 Local Properties of Morphisms: Affine Open Coverings of an Algebraic Variety
6.5 Veronese Morphism: Divisors and Linear Systems
6.5.1 Veronese morphism and consequences
6.5.2 Divisors and linear systems
Exercises
7. Products of Algebraic Varieties
7.1 Products of Affine Varieties
7.2 Products of Projective Varieties
7.2.1 Segre morphism and the product of projective spaces
7.2.2 Products of projective varieties
7.3 Products of Algebraic Varieties
7.4 Products of Morphisms
7.5 Diagonals, Graph of a Morphism and Fiber-Products
Exercises
8. Rational Maps of Algebraic Varieties
8.1 Rational and Birational Maps
8.1.1 Some properties and some examples of (bi)rational maps
8.2 Unirational and Rational Varieties
8.2.1 Stereographic projection of a rank-four quadric surface
8.2.2 Monoids
8.2.3 Blow-up of Pn at a point
8.2.4 Blow-ups and resolution of singularities
Exercises
9. Completeness of Projective Varieties
9.1 Complete Algebraic Var
9.2 The Main Theorem of Elimination Theory
9.2.1 Consequences of the main theorem of elimination theory
Exercises
10. Dimension of Algebraic Varieties
10.1 Dimension of an Algebraic Variety
10.2 Comparison on Various Definitions of “Dimension”
10.3 Dimension and Intersections
10.4 Complete Intersections
Exercises
11. Fiber-Dimension: Semicontinuity
11.1 Fibers of a Dominant Morphism
11.2 Semicontinuity
Exercises
12. Tangent Spaces: Smoothness of Algebraic Varieties
12.1 Tangent Space at a Point of an Affine Variety: Smoothness
12.2 Tangent Space at a Point of a Projective Variety: Smoothness
12.3 Zariski Tangent Space of an Algebraic Variety: Intrinsic Definition of Smoothness
Exercises
Solutions to Exercises
Bibliography
Index