An Introduction to Algebraic Geometry

Preface
Contents
1 Hilbert's Nullstellensatz
	1.1 Basic questions
	1.2 Gröbner bases
	1.3 Buchberger's criterion
	1.4 Proof of Hilbert's Nullstellensatz
2 The algebra-geometry dictionary
	2.1 Hilbert's strong Nullstellensatz
	2.2 Coordinate rings and morphisms
3 Noetherian rings and primary decomposition
	3.1 The ascending chain condition
	3.2 Primary decomposition
	3.3 Associated primes
	3.4 Filtration of modules with primes
4 Localization
	4.1 Fractions
	4.2 Some local properties
	4.3 Extended and contracted ideals
	4.4 Ideal theory of localizations
5 Rational functions and dimension
	5.1 The rational function field of a variety
	5.2 Dominant rational maps
	5.3 Appendix: The transcendence degree
6 Integral ring extensions and Krull dimension
	6.1 A Gröbner basis criterion for the dimension
	6.2 Integral ring extensions
	6.3 The lying-over theorem
	6.4 Krull dimension
7 Constructive ideal and module theory
	7.1 Syzgygies and applications
	7.2 Elimination and the kernel of a ring homomorphism
	7.3 Homomorphisms between modules
	7.4 Cokernel, image and kernel of an R-module homomorphism
8 Projective algebraic geometry
	8.1 The projective space
	8.2 The algebra-geometry dictionary in the projective case
	8.3 Hilbert's syzygy theorem
	8.4 The Hilbert function
9 Bézout's theorem
	9.1 Rational functions and regular functions on projective varieties
	9.2 Intersection multiplicities for plane curves
	9.3 Bézout's theorem for plane curves
	9.4 Bézout's theorem for the intersection with a hypersurface
10 Local rings and power series
	10.1 Local rings
	10.2 Formal power series and completions
	10.3 A lower bound on intersection multiplicities
	10.4 Mora division
	10.5 Differentiation and the tangent space
	10.6 Discrete valuation rings
11 Products and morphisms of projective varieties
	11.1 The Segre product
	11.2 Morphisms
	11.3 Dimension bounds
	11.4 The Veronese embeddings
	11.5 Morphisms from projective algebraic sets
	11.6 Semi-continuity of the fiber dimension
12 Resolution of curve singularities
	12.1 The blow-up
	12.2 Blow-up of smooth projective surfaces
13 Families of varieties
	13.1 The family of hypersurfaces
	13.2 Linear systems of plane curves
	13.3 The Grassmannian
	13.4 A glance at the Hilbert scheme
	13.5 `39`42`"613A``45`47`"603AHilb3t+1(P3)
	13.6 Ideals of leading terms from a Hilbert scheme point of view
14 Bertini's theorem and applications
	14.1 The dual variety
	14.2 The Riemann-Hurwitz formula
	14.3 Dynamical interpretation of intersection numbers
15 The geometric genus of a plane curve
	15.1 A bound on the number of singular points
	15.2 Existence of a Cremona resolution
	15.3 Rational curves
16 Riemann-Roch
	16.1 Divisors
	16.2 Rational differentials and canonical divisors
	16.3 Proof of the Riemann-Roch theorem
	16.4 Riemann's count
	16.5 The Clifford index and syzygies of canonical curves
A A glimpse of sheaves and cohomology
	A.1 Sheaves
	A.2 Cohomology
	A.3 Differentials and the adjunction sequence
	A.4 Intersection theory of curves on smooth projective surfaces
B Code for Macaulay2 computations
	B.1 Solutions to Exercises in Chapter 1, 2, 3, 4 and 6
	B.2 Solutions to Exercises in Chapter 8, 9, 10, 11 and 12
	B.3 The Computation in Section 13.5
	B.4 Solutions to Exercises in Chapter 14 and 16
References
Glossary
Index