Affine Algebraic Geometry : Geometry of Polynomial Rings

Contents
Preface
1. Introduction to Algebraic Geometry
	1.1 Review on basic results in commutative algebra
		1.1.1 Ring of quotients and local ring
		1.1.2 Spectrum of a ring and Zariski topology
		1.1.3 Irreducible decomposition of a topological space
		1.1.4 Prime ideal decomposition of radical ideals
		1.1.5 Generic point, closed point and Krull dimension
		1.1.6 Hilbert basis theorem
		1.1.7 Integral extension and Noether normalization lemma
		1.1.8 Lying-over theorem and Going-up theorem
		1.1.9 Krull dimension of affine domains
	1.2 Review on finitely generated field extensions
		1.2.1 Transcendence basis and transcendence degree
		1.2.2 Regular extension and separable extension
	1.3 Schemes and varieties
		1.3.1 Affine schemes of finite type and affine varieties
			1.3.1.1 Irreducible decomposition of an affine scheme of finite type
			1.3.1.2 Density of the set of closed points
			1.3.1.3 Affine varieties and function fields
			1.3.1.4 Structure sheaves
		1.3.2 Morphisms of affine schemes
			1.3.2.1 Intersection of affine open sets
			1.3.2.2 Open immersion and closed immersion
			1.3.2.3 Behavior of structure sheaves under a morphism
		1.3.3 Schemes and varieties
			1.3.3.1 Definition and examples of schemes
			1.3.3.2 Morphism of schemes
			1.3.3.3 Fiber products of schemes
			1.3.3.4 Separated schemes
			1.3.3.5 Rational maps of algebraic varieties
	1.4 Graded rings and projective schemes
		1.4.1 Graded rings and projective spectrums
		1.4.2 Projective schemes and projective varieties
			1.4.2.1 General properties of projective schemes
			1.4.2.2 Projective varieties
			1.4.2.3 Projective closure of an affine variety
	1.5 Normal varieties
		1.5.1 Discrete valuation rings and normal rings
		1.5.2 Normalization of affine domains
		1.5.3 Normal varieties and normalization of algebraic varieties
		1.5.4 Unique factorization domains
		1.5.5 Weil divisors and divisor class group
		1.5.6 Zariski’s main theorem
	1.6 Smooth varieties
		1.6.1 System of parameters and regular local ring
		1.6.2 Regular sequence and depth of a local ring
		1.6.3 Jacobian criterion
		1.6.4 Sheaf of differential 1-forms and canonical sheaf
	1.7 Divisors and linear systems
		1.7.1 Invertible sheaves
		1.7.2 Cartier divisors
		1.7.3 Linear systems
		1.7.4 D-dimension, Kodaira dimension and logarithmic Kodaira dimension
	1.8 Algebraic curves and surfaces
		1.8.1 Serre duality and Euler-Poincaré characteristic
		1.8.2 Riemann-Roch theorem for a curve
		1.8.3 Algebraic curves
		1.8.4 Intersection theory on algebraic surfaces
		1.8.5 Riemann-Roch theorem for surfaces
		1.8.6 Fibrations and relatively minimal models of surfaces
	1.9 Appendix to Chapter 1
		1.9.1 Primary decomposition of ideals
		1.9.2 Tensor products of algebras
			1.9.2.1 Construction
			1.9.2.2 Flat modules
		1.9.3 Inductive limits and projective limits
			1.9.3.1 Inductive limits
			1.9.3.2 Projective limits
			1.9.3.3 Ideal-adic completion
		1.9.4 Fiber products of schemes
		1.9.5 Reviews on sheaf theory
		1.9.6 Čech cohomology of sheaves of abelian groups
			1.9.6.1 Čech cohomology
			1.9.6.2 Coherent sheaf cohomologies over projective varieties
	1.10 Problems to Chapter 1
2. Geometry on Affine Surfaces
	2.1 Characterization of the affine plane
	2.2 Admissible data for an affine curve with one place at infinity
		2.2.1 Euclidean transformation associated with admissible data
		2.2.2 (e, i)-transformation associated with admissible data
		2.2.3 Irreducible affine curves with one-place at infinity
		2.2.4 Abhyankar-Moh-Suzuki theorem
		2.2.5 Theorem of Gutwirth and pathological A1-fibrations
		2.2.6 Abhyankar-Moh problem on embedded lines in positive characteristic
	2.3 Automorphism theorem of the affine plane
		2.3.1 Linear pencils of rational curves and field generators
		2.3.2 Proof of automorphism theorem by Jung and van der Kulk
	2.4 Algebraic group actions on the affine plane
		2.4.1 Algebraic groups, actions and quotient spaces
		2.4.2 Finite subgroups of Aut k[x, y]
		2.4.3 Finite group actions and invariants
		2.4.4 Quotient singularities on surfaces
	2.5 Birational automorphisms of rational surfaces
		2.5.1 Noether factorization theorem
	2.6 Boundary divisors of affine surfaces
		2.6.1 Quantitative criterion of SNC divisors
		2.6.2 Shift transformation on the boundary divisor
		2.6.3 Theorem of Ramanujam-Morrow
	2.7 Appendix to Chapter 2
		2.7.1 Unramified morphism
		2.7.2 Étale coverings
		2.7.3 Riemann-Hurwitz formula for curves
		2.7.4 Inverse and direct images of divisors and the projection formula
		2.7.5 Amalgamated product of two groups
		2.7.6 Quotient varieties by finite group actions and ramification of the quotient morphism
	2.8 Problems to Chapter 2
3. Geometry and Topology of Polynomial Rings — Motivated by the Jacobian Problem
	3.1 Plane-like affine surfaces
		3.1.1 Simply connected algebraic varieties
		3.1.2 Unit group, unit rank and independence of boundary divisors
		3.1.3 Gizatullin surfaces and affine pseudo-planes
		3.1.4 Affine pseudo-planes — more properties
		3.1.5 tom Dieck construction of affine pseudo-planes
		3.1.6 Platonic A1∗-fiber spaces
		3.1.7 Homology planes
	3.2 Jacobian conjecture and related results
		3.2.1 Jacobian conjecture and its variants
		3.2.2 Partial affirmative answers
	3.3 Generalized Jacobian conjecture — affirmative cases
		3.3.1 Results in arbitrary dimension
		3.3.2 Results for surfaces
			3.3.2.1 Surfaces having A1-fibrations
			3.3.2.2 Surfaces having A1∗-fibrations
			3.3.2.3 Case of κ = 1
	3.4 Generalized Jacobian conjecture for various cases
		3.4.1 Case of Q-homology planes of κ = −∞
		3.4.2 Counterexamples
	3.5 Appendix to Chapter 3
		3.5.1 Makar-Limanov invariant
		3.5.2 The fundamental group at infinity
		3.5.3 Algebraic surfaces and log Kodaira dimension
		3.5.4 Logarithmic ramification formula
	3.6 Problems to Chapter 3
4. Postscript
	4.1 AMS theorem and thereafter
	4.2 Suzuki-Zaidenberg formula
	4.3 Cancellation problems
Bibliography
Index