Algebraic Curves and Riemann Surfaces for Undergraduates: The Theory of the Donut

Preface
Contents
List of Symbols
1 Introduction
1.1 The Theory of the Circle
1.1.1 Pythagorean Triples
1.1.2 The Circular Functions
The Isomorphism Theorem for the Circle
1.1.3 The Theory of the Donut, in a Nutshell
1.2 Overview of the Book
1.2.1 Part I: Algebraic Curves
Affine and Projective Curves
Intersections of Curves
Elliptic Curves
1.2.2 Part II: Riemann Surfaces
Three Kinds of Surfaces
Analytic Functions
Real Analysis, Complex Analysis, and Path Integrals
Finally, Riemann Surfaces
1.2.3 Part III: Curves and Surfaces
1.3 Preliminaries, and Some Notation
Part I Algebraic Curves
2 Algebra
2.1 Polynomials and Power Series
2.1.1 The Category of Rings
Simplifying Notation
Derived Properties
2.1.2 Back to Formal Power Series
Infinite Sums
Several Variables
2.1.3 More on Polynomials
The Degree of a Polynomial
Polynomial Substitution
2.2 Unique Factorisation
2.2.1 Divisibility in Integral Domains
Divisibility in R[x]
2.2.2 Unique Factorisation Domains
Multisets
Unique Factorisation
2.2.3 Unique Factorisation in Polynomial Rings
One Variable
Interlude: Algebraically Closed Fields
Gauss's Lemmas
2.3 Groups
2.3.1 The Category of Groups
Subgroups
Group Homomorphisms
2.3.2 Quotient Groups
2.3.3 Cyclic Groups
The Characteristic of a Ring
2.3.4 The Symmetric Group
2.4 Linear Algebra Over Integral Domains
2.4.1 Matrices, Linear Spaces, and Linear Maps
Linear Spaces
Invertible and Nonsingular Matrices
2.4.2 Dimension and Complements
2.4.3 The Determinant
The Effect of Row Operations
Polynomial Substitution
2.4.4 Detecting Singularity
2.5 Further Exercises
3 Affine Space
3.1 Definition of Hypersurfaces
3.2 The Resultant
3.2.1 The Sylvester Matrix
3.2.2 The Resultant, Common Roots, and More Variables
Adding More Variables
3.2.3 The Resultant is a Linear Combination
3.3 Study's Lemma
3.3.1 Proof of Study's Lemma
3.4 Affine Lines and Rational Parameterisations
3.4.1 Affine Lines
3.4.2 Rational Parameterisations
3.5 Further Exercises
4 Projective Space
4.1 Homogeneous Polynomials
A Characterisation of Homogeneity
4.2 Projective Space
4.3 Projective Lines and Maps
4.3.1 Projective Maps
4.4 Embedding Affine Space into Projective Space
Vertical and Horizontal Projective Lines
Affine Cover and the Riemann Sphere
Algebraic Subsets of the Projective Line
4.5 Changes of Coordinates
4.5.1 Change of Variable
4.5.2 Four Point Lemma
4.6 Spaces of Curves
4.6.1 The Dual Plane
Principle of Duality
4.6.2 Desargues' Theorem
4.7 Products of Projective Spaces
4.8 Further Exercises
5 Tangents
5.1 Introduction: Affine Tangents and Intersections with Lines
5.1.1 Intersection Multiplicities
5.1.2 Homogeneous Coordinates
5.2 Formal Partial Derivatives
5.2.1 Properties of Derivatives
The Chain Rule
Euler's Relation
Taylor Expansions
5.2.2 The Discriminant
5.3 Higher Order Tangents
The Affine Higher Order Tangent
5.3.1 The Moduli Space of Tangents
5.3.2 Invariance of the Higher Order Tangent
5.4 The Intersection of a Line with a Curve
5.4.1 Definition of Intersection Multiplicity
Bézout for a Line
5.4.2 Invariance of Multiplicity of Intersection with a Line
Affine Calculations
5.4.3 Tangents and Intersections with Lines
Defining Multiplicities Using Tangents
5.4.4 Simple Intersections Are the Norm
5.5 Further Exercises
6 Bézout's Theorem
6.1 A First Look at the Intersection of Curves
6.1.1 The Resultant of Homogeneous Polynomials Is Homogeneous
6.1.2 A Weak Version of Bézout's Theorem
A Naïve Definition of Intersection Multiplicity
6.2 The Homogeneous Resultant
6.2.1 Main Property of the Homogeneous Resultant
6.3 Multiplicity of Intersection and Bézout's Theorem
6.3.1 Coding Lines in ps: [/EMC pdfmark [/Subtype /Span /ActualText (double struck upper P squared times double struck upper P squared) /StPNE pdfmark [/StBMC pdfmarkP2P2ps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark
6.3.2 The Resultant of the General Intersection Polynomials
Bihomogeneity of Rf,g
The Structure of the Hypersurface Defined by Rf,g
6.3.3 Intersection Multiplicity and Bézout's Theorem
The Intersection Multiset
6.3.4 Geometric Invariance
6.4 Coincidence with Earlier Definitions
6.4.1 Using the Family of Vertical Lines
6.4.2 Intersecting Lines
6.5 Categoricity of Multiplicity of Intersection
6.5.1 Symmetry
6.5.2 Products
6.5.3 Infinite Multiplicities
6.5.4 Shifts
6.5.5 Categoricity of Multiplicity of Intersection
6.6 Affine Calculations
6.7 Multiplicities, Orders and Tangents
6.8 Further Exercises
7 The Elliptic Group
7.1 Flexes
7.1.1 Flexes and the Second Order Tangent
7.1.2 The Hessian
7.2 The Group Operation on a Nonsingular Cubic Curve
7.2.1 The Complement Curve
7.2.2 Associativity of the Group Operation
7.3 Normal Forms for Nonsingular Cubics
7.3.1 Explicit Calculations of the Group Operation
7.4 Further Exercises
Part II Riemann Surfaces
8 Quasi-Euclidean Spaces
8.1 Topology of ps: [/EMC pdfmark [/Subtype /Span /ActualText (double struck upper R Superscript n) /StPNE pdfmark [/StBMC pdfmarkRnps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark
8.2 Manifolds
Examples
8.2.1 Topology of Pre-manifolds
8.2.2 Subspaces
8.2.3 The Hausdorff Property
8.2.4 Topological Countability
8.2.5 Manifolds
8.2.6 Spaces and Continuity
8.3 Compactness
8.3.1 Closed Sets
8.3.2 Sequences and Limits
8.3.3 Interlude: Completeness
8.3.4 Compactness in Euclidean Space
Uniform Continuity
Distances from Sets
8.4 Quotients by Discrete Subgroups
8.5 Further Exercises
9 Connectedness, Smooth and Simple
9.1 Connectedness, Path and Simple
9.1.1 Homotopy; Simple Connectedness
9.2 Lifting Maps
9.2.1 The Winding Number
9.3 Differentiability: A Reminder
9.3.1 Mean Value Inequalities
9.3.2 Partial Derivatives
9.3.3 Inverse Functions
9.3.4 Second Derivatives
9.4 Differentiable Manifolds
9.5 Partitions of Unity
9.5.1 Proof of Theorem 9.66
9.6 Differentiable Connectedness
9.6.1 Piecewise Smooth Paths
9.7 Further Exercises
10 Path Integrals
10.1 Integrating Forms Along Paths
Definition of the Integral
Properties of the Integral
10.1.1 The Length of a Path
10.2 Integrating Along Smooth Paths
10.2.1 Linear Forms
10.2.2 Relating the General and Familiar Integrals
The Fundamental Theorem of Calculus
Derivative of an Integral Depending on a Parameter
10.3 Integrating Vector Fields
10.3.1 Conservative Vector Fields
10.3.2 The Winding Number Revisited
10.4 Symmetric Vector Fields
10.4.1 Missing a Point
10.5 Further Exercises
11 Complex Differentiation
11.1 Complex Derivatives and Integrals
11.1.1 Complex Integrals
11.2 Cauchy's Integral Formula
Continuously Differentiable Functions and Primitives
11.2.1 Winding Numbers in the Complex Plane
The Integral Formula
11.3 Uniform Convergence and Power Series
11.3.1 Absolute Convergence
Rearrangements
11.3.2 Uniform Convergence
Convergence on Compact Sets
11.3.3 Power Series
11.4 Analytic Functions
11.4.1 Differentiating Power Series
11.4.2 The Exponential and Trigonometric Functions
11.4.3 Continuously Differentiable Functions Are Analytic
11.5 Morera, Weierstrass, Liouville
11.5.1 Liouville's Theorem
11.6 Further Exercises
12 Riemann Surfaces
12.1 Holomorphic Surfaces
12.1.1 Meromorphic Functions
The Meromorphic Conjugate
12.2 The Open Mapping Theorem
12.2.1 The Calculus of Residues
12.2.2 The Continuity of Roots of Polynomials
12.2.3 Open Mappings and Inverse Functions
Consequences for Riemann Surfaces
12.3 Compact Riemann Surfaces
12.4 Riemann Surfaces for the Logarithm and Roots
12.4.1 The Logarithm
12.4.2 The Surface for the nth Root
The Shift on
12.5 Analytic Continuation
12.6 Differential Forms on Surfaces
12.6.1 Pull-Backs of Meromorphic Forms
12.6.2 Quotients of Forms
The Differential of a Meromorphic Function
12.6.3 Integration of Holomorphic Forms
12.7 Further Exercises
Part III Curves and Surfaces
13 Curves Are Surfaces
13.1 The Implicit Function Theorem
13.2 Nonsingular Curves Are Riemann Surfaces
13.2.1 Vertical Parameterisations
13.2.2 An Atlas for the Nonsingular Part of a Curve
13.2.3 Rational Functions on Curves
13.2.4 Lifting Paths to Curves
Algebraic Curves Are Connected
13.3 Intersections with Lines, Revisited
13.3.1 Continuous Intersection Multiplicities
13.3.2 Finding Intersection Points
13.3.3 Finding Intersecting Lines
13.3.4 An Application to Elliptic Curves
13.4 Further Exercises
14 Elliptic Functions and the Isomorphism Theorem
14.1 Elliptic Functions
14.1.1 The Weierstrass Function
Definition of
Is an Elliptic Function
Inverse Images of Points
14.1.2 The Differential Equation for
14.2 The Curve E and the Isomorphism Theorem
14.2.1 The Isomorphism Theorem
14.3 Inversion
14.3.1 A Non-vanishing Form on a Nonsingular Cubic
14.3.2 Working After the Fact
14.3.3 Invariance of the Non-vanishing Holomorphic Form
14.3.4 Proof of the Inversion Theorem
14.4 Further Exercises
15 Puiseux Theory
15.1 Fractional Power Series and Their Holomorphic Functions
15.1.1 Formal and Informal Power Series
Germs
15.1.2 Substitutions into Power Series
15.1.3 Fractional Power Series
15.1.4 The Holomorphic Function Defined by a Fractional Power Series
The Induced Function on the Root Surface
The Shift of a Fractional Power Series
15.2 Parameterisations of a Curve
15.2.1 n-Fold Parameterisations
15.2.2 Fractional Parameterisations
15.2.3 Existence of Parameterisations
15.3 Branches and Places
15.3.1 Central Places
15.3.2 Branches of a Curve
15.4 Puiseux Expansions and Factorisation into Places
15.4.1 Puiseux Expansions
15.4.2 The Implicit Definition of a Place
A Factorisation of the Defining Polynomial
15.5 Intersection Multiplicities Using Places
15.5.1 Intersections of Curves and Places
15.5.2 Intersections of Curves
15.5.3 Orders and Tangents of Places
15.5.4 Some Nifty Consequences
Intersections with Shifted Curves
Intersection Multiplicity, Orders, and Shared Tangents
15.6 Further Exercises
16 A Brief History of Elliptic Functions
16.1 A History of Circles and Ellipses
Interlude: The Circular Functions
Inverting Elliptic Integrals
Bibliography
Index