An Undergraduate Primer in Algebraic Geometry

Preface
Contents
1 Affine and Projective Algebraic Sets
	1.1 Affine Algebraic Sets
	1.2 Projective Spaces
	1.3 Graded Rings
	1.4 Projective Algebraic Sets
	1.5 Projective Closure of Affine Sets
	1.6 Examples
		1.6.1 Points
		1.6.2 Projective Subspaces
		1.6.3 Affine Subspaces
		1.6.4 Hypersurfaces
		1.6.5 Divisors
		1.6.6 Product Topology
	1.7 Solutions of Some Exercises
2 Basic Notions of Elimination Theory and Applications
	2.1 The Resultant of Two Polynomials
	2.2 The Intersection of Two Plane Curves
	2.3 Kronecker Elimination Method: One Variable
	2.4 Kronecker Elimination Method: More Variables
	2.5 Hilbert Nullstellensatz
	2.6 Solutions of Some Exercises
3 Zariski Closed Subsets and Ideals in the Polynomials Ring
	3.1 Ideals and Coordinate Rings
	3.2 Examples
		3.2.1 Maximal Ideals
		3.2.2 The Twisted Cubic
		3.2.3 Cones
	3.3 Solutions of Some Exercises
4 Some Topological Properties
	4.1 Irreducible Sets
	4.2 Noetherian Spaces
	4.3 Topological Dimension
	4.4 Solutions of Some Exercises
5 Regular and Rational Functions
	5.1 Regular Functions
	5.2 Rational Functions
	5.3 Local Rings
	5.4 Integral Elements over a Ring
	5.5 Subvarieties and Their Local Rings
	5.6 Product of Affine Varieties
	5.7 Solutions of Some Exercises
6 Morphisms
	6.1 The Definition of Morphism
	6.2 Which Maps Are Morphisms
	6.3 Affine Varieties
	6.4 The Veronese Morphism
	6.5 Solutions of Some Exercises
7 Rational Maps
	7.1 Definition of Rational Maps and Basic Properties
	7.2 Birational Models of Quasi-projective Varieties
	7.3 Unirational and Rational Varieties
	7.4 Solutions of Some Exercises
8 Product of Varieties
	8.1 Segre Varieties
	8.2 Products
	8.3 The Blow–up
	8.4 Solutions of Some Exercises
9 More on Elimination Theory
	9.1 The Fundamental Theorem of Elimination Theory
	9.2 Morphisms on Projective Varieties Are Closed
	9.3 Solutions of Some Exercises
10 Finite Morphisms
	10.1 Definitions and Basic Results
	10.2 Projections and Noether's Normalization Theorem
	10.3 Normal Varieties and Normalization
	10.4 Ramification
	10.5 Solutions of Some Exercises
11 Dimension
	11.1 Characterization of Hypersurfaces
	11.2 Intersection with Hypersurfaces
	11.3 Morphisms and Dimension
	11.4 Elimination Theory Again
	11.5 Solutions of Some Exercises
12 The Cayley Form
	12.1 Definition of the Cayley Form
	12.2 The Degree of a Variety
	12.3 The Cayley Form and Equations of a Variety
	12.4 Cycles and Their Cayley Forms
	12.5 Solutions of Some Exercises
13 Grassmannians
	13.1 Plücker Coordinates
	13.2 Grassmann Varieties
	13.3 Solutions of Some Exercises
14 Smooth and Singular Points
	14.1 Basic Definitions
	14.2 Some Properties of Smooth Points
		14.2.1 Regular Rings
		14.2.2 System of Parameters
		14.2.3 Auslander–Buchsbaum Theorem
		14.2.4 Local Equations of a Subvariety
	14.3 Smooth Curves and Finite Maps
	14.4 A Criterion for a Map to Be an Isomorphism
	14.5 Solutions of Some Exercises
15 Power Series
	15.1 Formal Power Series
	15.2 Congruences, Substitution and Derivatives
		15.2.1 Conguences
		15.2.2 Substitution
		15.2.3 Derivatives
	15.3 Fractional Power Series
	15.4 Solutions of Some Exercises
16 Affine Plane Curves
	16.1 Multiple Points and Principal Tangent Lines
	16.2 Parametrizations and Branches of a Curve
	16.3 Intersections of Affine Curves
		16.3.1 Intersection Multiplicity and Resultants
		16.3.2 Order of a Curve at a Branch and Intersection Multiplicities
		16.3.3 More Properties of Branches and of Intersection Multiplicity
		16.3.4 Further Interpretation of the Intersection Multiplicity
	16.4 Solutions of Some Exercises
17 Projective Plane Curves
	17.1 Some Generalities
		17.1.1 Recalling Some Basic Definitions
		17.1.2 The Bézout Theorem
		17.1.3 Linear Systems
	17.2 M. Noether's Af+Bg Theorem
	17.3 Applications of the Af+Bg Theorem
		17.3.1 Pascal's and Pappo's Theorems
		17.3.2 The Group Law on a Smooth Cubic
	17.4 Solutions of Some Exercises
18 Resolution of Singularities of Curves
	18.1 The Case of Ordinary Singularities
	18.2 Reduction to Ordinary Singularities
		18.2.1 Statement of the Main Theorem
		18.2.2 Standard Quadratic Transformations
		18.2.3 Transformation of a Curve via a Standard Quadratic Transformation
		18.2.4 Proof of the Main Theorem
19 Divisors, Linear Equivalence, Linear Series
	19.1 Divisors
	19.2 Linear Equivalence
	19.3 Fibres of a Morphism
	19.4 Linear Series
	19.5 Linear Series and Projective Morphisms
	19.6 Adjoint Curves
	19.7 Linear Systems of Plane Curves and Linear Series
	19.8 Solutions of Some Exercises
20 The Riemann–Roch Theorem
	20.1 The Riemann–Roch Theorem
	20.2 Consequences of the Riemann–Roch Theorem
	20.3 Differentials
		20.3.1 Algebraic Background
		20.3.2 Differentials and Canonical Divisors
		20.3.3 The Riemann–Hurwitz Theorem
	20.4 Solutions of Some Exercises
Appendix  References
Index