- #1
- 22,183
- 3,322
- Author: Rick Miranda
- Title: Algebraic Curves and Riemann Surfaces
- Amazon Link: https://www.amazon.com/dp/0821802682/?tag=pfamazon01-20
- Prerequisities: Complex Analysis, Differential Geometry, Abstract Algebra
- Level: Grad
Table of Contents:
Code:
[LIST]
[*] Preface
[*] Riemann Surfaces: Basic Definitions
[LIST]
[*] Complex Charts and Complex Structures
[LIST]
[*] Complex Charts
[*] Complex Atlases
[*] The Definition of a Riemann Surface
[*] Real 2-Manifolds
[*] The Genus of a Compact Riemann Surface
[*] Complex Manifolds
[*] Problems
[/LIST]
[*] First Examples of Riemann Surfaces
[LIST]
[*] A Remark on Defining Riemann Surfaces
[*] The Projective Line
[*] Complex Tori
[*] Graphs of Holomorphic Functions
[*] Smooth Affine Plane Curves
[*] Problems
[/LIST]
[*] Projective Curves
[LIST]
[*] The Projective Plane P^2
[*] Smooth Projective Plane Curves
[*] Higher-Dimensional Projective Spaces
[*] Complete Intersections
[*] Local Complete Intersections
[*] Problems
[/LIST]
[*] Further Reading
[/LIST]
[*] Functions and Maps
[LIST]
[*] Functions on Riemann Surfaces
[LIST]
[*] Holomorphic Functions
[*] Singularities of Functions; Meromorphic Functions
[*] Laurent Series
[*] The Order of a Meromorphic Function at a Point
[*] C^\infty Functions
[*] Harmonic Functions
[*] Theorems Inherited from One Complex Variable
[*] Problems
[/LIST]
[*] Examples of Meromorphic Functions
[LIST]
[*] Meromorphic Functions on the Riemann Sphere
[*] Meromorphic Functions on the Projective Line
[*] Meromorphic Functions on a Complex Torus
[*] Meromorphic Functions on Smooth Plane Curves
[*] Smooth Projective Curves
[*] Problems
[/LIST]
[*] Holomorphic Maps Between Riemann Surfaces
[LIST]
[*] The Definition of a Holomorphic Map
[*] Isomorphisms and Automorphisms
[*] Easy Theorems about Holomorphic Maps
[*] Meromorphic Functions and Holomorphic Maps to the Riemann Sphere
[*] Meromorphic Functions on a Complex Torus, Again
[*] Problems
[/LIST]
[*] Global Properties of Holomorphic Maps
[LIST]
[*] Local Normal Form and Multiplicity
[*] The Degree of a Holomorphic Map between Compact Riemann Surfaces
[*] The Sum of the Orders of a Meromorphic Function
[*] Meromorphic Functions on a Complex Torus, Yet Again
[*] The Euler Number of a Compact Surface
[*] Hurwitz's Formula
[*] Problems
[/LIST]
[*] Further Reading
[/LIST]
[*] More Examples of Riemann Surfaces
[LIST]
[*] More Elementary Examples of Riemann Surfaces
[LIST]
[*] Lines and Conics
[*] Glueing Together Riemann Surfaces
[*] Hyperelliptic Riemann Surfaces
[*] Meromorphic Functions on Hyperelliptic Riemann Surfaces
[*] Maps Between Complex Tori
[*] Problems
[/LIST]
[*] Less Elementary Examples of Riemann Surfaces
[LIST]
[*] Plugging Holes in Riemann Surfaces
[*] Nodes of a Plane Curve
[*] Resolving a Node of a Plane Curve
[*] The Genus of a Projective Plane Curve with Nodes
[*] Resolving Monomial Singularities
[*] Cyclic Coverings of the Line
[*] Problems
[/LIST]
[*] Group Actions on Riemann Surfaces
[LIST]
[*] Finite Group Actions
[*] Stabilizer Subgroups
[*] The Quotient Riemann Surface
[*] Ramification of the Quotient Map
[*] Hurwitz's Theorem on Automorphisms
[*] Infinite Groups
[*] Problems
[/LIST]
[*] Monodromy
[LIST]
[*] Covering Spaces and the Fundamental Group
[*] The Monodromy of a Finite Covering
[*] The Monodromy of a Holomorphic Map
[*] Coverings via Monodromy Representations
[*] Holomorphic Maps via Monodromy Representations
[*] Holomorphic Maps to ? 1
[*] Hyperelliptic Surfaces
[*] Problems
[/LIST]
[*] Basic Projective Geometry
[LIST]
[*] Homogeneous Coordinates and Polynomials
[*] Projective Algebraic Sets
[*] Linear Subspaces
[*] The Ideal of a Projective Algebraic Set
[*] Linear Automorphisms and Changing Coordinates
[*] Projections
[*] Secant and Tangent Lines
[*] Projecting Projectlye Curves
[*] Problems
[/LIST]
[*] Further Reading
[/LIST]
[*] Integration on Riemann Surfaces
[LIST]
[*] Differential Forms
[LIST]
[*] Holomorphic 1-Forms
[*] Meromorphic 1-Forms
[*] Defining Meromorphic Functions and Forms with a Formula
[*] Using dz and d\bar{z}
[*] C^\infty 1-Forms
[*] 1-Forms of Type (1,0) and (0, 1)
[*] C^\infty 2-Forms
[*] Problems
[/LIST]
[*] Operations on Differential Forms
[LIST]
[*] Multiplication of 1-Forms by Functions
[*] Differentials of Functions
[*] The Wedge Product of Two 1-Forms
[*] Differentiating 1-Forms
[*] Pulling Back Differential Forms
[*] Some Notation
[*] The Poincar and Dolbeault Lemmas
[*] Problems
[/LIST]
[*] Integration on a Riemann Surface
[LIST]
[*] Paths
[*] Integration of i-Forms Along Paths
[*] Chains and Integration Along Chains
[*] The Residue of a Meromorphic 1-Form
[*] Integration of 2-Forms
[*] Stoke's Theorem
[*] The Residue Theorem
[*] Homotopy
[*] Homology
[*] Problems
[/LIST]
[*] Further Reading
[/LIST]
[*] Divisors and Meromorphic Functions
[LIST]
[*] Divisors
[LIST]
[*] The Definition of a Divisor
[*] The Degree of a Divisor on a Compact Riemann Surface
[*] The Divisor of a Meromorphic Function: Principal Divisors
[*] The Divisor of a Meromorphic 1-Form: Canonical Divisors
[*] The Degree of a Canonical Divisor on a Compact Riemann Surface
[*] The Boundary Divisor of a Chain
[*] The Inverse Image Divisor of a Holomorphic Map
[*] The Ramification and Branch Divisor of a Holomorphic Map
[*] Intersection Divisors on a Smooth Projective Curve
[*] The Partial Ordering on Divisors
[*] Problems
[/LIST]
[*] Linear Equivalence of Divisors
[LIST]
[*] The Definition of Linear Equivalence
[*] Linear Equivalence for Divisors on the Riemann Sphere
[*] Principal Divisors on a Complex Torus
[*] The Degree of a Smooth Projective Curve
[*] Bezout's Theorem for Smooth Projective Plane Curves
[*] Plucker's Formula
[*] Problems
[/LIST]
[*] Spaces of Functions and Forms Associated to a Divisor
[LIST]
[*] The Definition of the Space L(D)
[*] Complete Linear Systems of Divisors
[*] Isomorphislns between L(D)'s under Linear Equivalence
[*] The Definition of the Space L^{(1)}(D)
[*] The Isomorphism between L^{(1)}(D) and L(D + K)
[*] Computation of L(D) for the Riemann Sphere
[*] Computation of L(D) for a Complex Torus
[*] A Bound on the Dimension of L(D)
[*] Problems
[/LIST]
[*] Divisors and Maps to Projective Space
[LIST]
[*] Holomorphic Maps to Projective Space
[*] Maps to Projective Space Given By Meromorphic Functions
[*] The Linear System of a Holomorphic Map
[*] Base Points of Linear Systems
[*] The Hyperplane Divisor of a Holomorphic Map to P^n
[*] Defining a Holomorphic Map via a Linear System
[*] Removing the Base Points
[*] Criteria for \phi_D to be an Embedding
[*] The Degree of the Image and of the Map
[*] Rational and Elliptic Normal Curves
[*] Working Without Coordinates
[*] Problems
[/LIST]
[*] Further Reading
[/LIST]
[*] Algebraic Curves and the Riemann-Roch Theorem
[LIST]
[*] Algebraic Curves
[LIST]
[*] Separating Points and Tangents
[*] Constructing Functions with Specified Laurent Tails
[*] The Transcendence Degree of the Function Field M(X)
[*] Computing the Function Field M(X)
[*] Problems
[/LIST]
[*] Laurent Tail Divisors
[LIST]
[*] Definition of Laurent Tail Divisors
[*] Mittag-Lefiter Problems and H^1(D)
[*] Comparing H^1 Spaces
[*] The Finite-Dimensionality of H^1(D)
[*] Problems
[/LIST]
[*] The Riemann-Roch Theorem and Serre Duality
[LIST]
[*] The Riemann-Roch Theorem I
[*] The Residue Map
[*] Serre Duality
[*] The Equality of the Three Genera
[*] The Riemann-Roch Theorem II
[*] Problems
[/LIST]
[*] Further Reading
[/LIST]
[*] Applications of Riemann-Roch
[LIST]
[*] First Applications of Riemann-Roch
[LIST]
[*] How Riemann-Roch implies Algebraicity
[*] Criterion for a Divisor to be Very Ample
[*] Every Algebraic Curve is Projective
[*] Curves of Genus Zero are Isomorphic to the Riemann Sphere
[*] Curves of Genus One are Cubic Plane Curves
[*] Curves of Genus One are Complex Tori
[*] Curves of Genus Two are Hyperelliptic
[*] Clifford's Theorem
[*] The Canonical System is Base-Point-Free
[*] The Existence of Meromorphic 1-Forms.
[*] Problems
[/LIST]
[*] The Canonical Map
[LIST]
[*] The Canonical Map for a Curve of Genus at Least Three
[*] The Canonical Map for a Hyperelliptic Curve
[*] Finding Equations for Smooth Projective Curves
[*] Classification of Curves of Genus Three
[*] Classification of Curves of Genus Four
[*] The Geometric Form of Riemann-Roch
[*] Classification of Curves of Genus Five
[*] The Space L(D) for a General Divisor
[*] A Few Words on Counting Parameters
[*] Riemann's Count of 3g - 3 Parameters for Curves of Genus g
[*] Problems
[/LIST]
[*] The Degree of Projective Curves
[LIST]
[*] The Minimal Degree
[*] Rational Normal Curves
[*] Tangent Hyperplanes
[*] Flexes and Bitangents
[*] Monodromy of the Hyperplane Divisors
[*] The Surjectivity of the Monodromy
[*] The General Position Lemma
[*] Points Imposing Conditions on Hypersurfaces
[*] Castelnuovo's Bound
[*] Curves of Maximal Genus
[*] Problems
[/LIST]
[*] Inflection Points and Weierstrass Points
[LIST]
[*] Gap Numbers and Inflection Points of a Linear System
[*] The Wronskian Criterion
[*] Higher-order Differentials
[*] The Number of Inflection Points
[*] Flex Points of Smooth Plane Curve
[*] Weierstrass Points
[*] Problems
[/LIST]
[*] Further Reading
[/LIST]
[*] Abel's Theorem
[LIST]
[*] Homology, Periods, and the Jacobian
[LIST]
[*] The First Homology Group
[*] The Standard Identified Polygon
[*] Periods of 1-Forms
[*] The Jacobian of a Compact Riemann Surface
[*] Problems
[/LIST]
[*] The Abel-Jacobi Map
[LIST]
[*] The Abel-Jacobi Map A on X
[*] The Extension of A to Divisors
[*] Independence of the Base Point
[*] Statement of Abel's Theorem
[*] Problems
[/LIST]
[*] Trace Operations
[LIST]
[*] The Trace of a Function
[*] The Trace of a 1-Form
[*] The Residue of a Trace
[*] An Algebraic Proof of the Residue Theorem
[*] Integration of a Trace
[*] Proof of Necessity in Abel's Theorem
[*] Problems
[/LIST]
[*] Proof of Sufficiency in Abel's Theorem
[LIST]
[*] Lemmas Concerning Periods
[*] The Proof of Sufficiency
[*] Riemann's Bilinear Relations
[*] The Jacobian and the Picard Group
[*] Problems
[/LIST]
[*] Abel's Theorem for Curves of Genus One
[LIST]
[*] The Abel-Jacobi Map is an Embedding
[*] Every Curve of Genus One is a Complex Torus
[*] The Group Law on a Smooth Projective Plane Cubic
[*] Problems
[/LIST]
[*] Further Reading
[/LIST]
[*] Sheaves and Cech Cohomology
[LIST]
[*] Presheaves and Sheaves
[LIST]
[*] Presheaves
[*] Examples of Presheaves
[*] The Sheaf Axiom
[*] Locally Constant Sheaves
[*] Skyscraper Sheaves
[*] Global Sections on Compact Riemann Surfaces
[*] Restriction to an Open Subset
[*] Problems
[/LIST]
[*] Sheaf Maps
[LIST]
[*] Definition of a Map between Sheaves
[*] Inclusion Maps
[*] Differentiation Maps
[*] Restriction or Evaluation Maps
[*] Multiplication Maps
[*] Truncation Maps
[*] The Exponential Map
[*] The Kernel of a Sheaf Map
[*] 1-1 and Onto Sheaf Maps
[*] Short Exact Sequences of Sheaves
[*] Exact Sequences of Sheaves
[*] Sheaf Isomorphisms
[*] Using Sheaves to Define the Category
[*] Problems
[/LIST]
[*] Cech Cohomology of Sheaves
[LIST]
[*] Cech Cochains
[*] Cech Cochain Complexes
[*] Cohomology with respect to a Cover
[*] Refinements
[*] Cech Cohomology Groups
[*] The Connecting Homomorphism
[*] The Long Exact Sequence of Cohomology
[*] Problems
[/LIST]
[*] Cohomology Computations
[LIST]
[*] The Vanishing of H^1 for C^\infty Sheaves
[*] The Vanishing of H^1 for Skyscraper Sheaves
[*] Cohomology of Locally Constant Sheaves
[*] The Vanishing of H^1(X,O_X[D])
[*] De Rham Cohomology
[*] Dolbeault Cohomology
[*] Problems
[/LIST]
[*] Further Reading
[/LIST]
[*] Algebraic Sheaves
[LIST]
[*] Algebraic Sheaves of Functions and Forms
[LIST]
[*] Algebraic Curves
[*] Algebraic Sheaves of Functions
[*] Algebraic Sheaves of Forms
[*] The Zariski Topology
[*] Problems
[/LIST]
[*] Zariski Cohomology
[LIST]
[*] The Vanishing of H^1(X_{Zar}, F) for a Constant Sheaf
[*] The Interpretation of H^1(D)
[*] GAGA Theorems
[*] Further Computations
[*] The Zero Mean Theorem
[*] The High Road to Abel's Theorem
[*] Problems
[/LIST]
[*] Further Reading
[/LIST]
[*] Invertible Sheaves, Line Bundles, and H^1
[LIST]
[*] Invertible Sheaves
[LIST]
[*] Sheaves of O-Modules
[*] Definition of an Invertible Sheaf
[*] Invertible Sheaves associated to Divisors
[*] The Tensor Product of Invertible Sheaves
[*] The Inverse of an Invertible Sheaf
[*] The Group of Isomorphism Classes of Invertible Sheaves
[*] Problems
[/LIST]
[*] Line Bundles
[LIST]
[*] The Definition of a Line Bundle
[*] The Tautological Line Bundle for a Map to P^n
[*] Line Bundle Homomorphisms
[*] Defining a Line Bundle via Transition Functions
[*] The Invertible Sheaf of Regular Sections of a Line Bundle
[*] Sections of the Tangent Bundle and Tangent Vector Fields
[*] Rational Sections of a Line Bundle
[*] The Divisor of a Rational Section
[*] Problems
[/LIST]
[*] Avatars of the Picard Group
[LIST]
[*] Divisors Modulo Linear Equivalence and Cocycles
[*] Invertible Sheaves Modulo Isomorphism
[*] Line Bundles Modulo Isomorphism
[*] The Jacobian
[*] Problems
[/LIST]
[*] H^1 as a Classifying Space
[LIST]
[*] Why H^1(O*) Classifies Invertible Sheaves and Line Bundles
[*] Locally Trivial Structures
[*] A General Principle Regarding H^1
[*] Cyclic Unbranched Coverings
[*] Extensions of Invertible Sheaves
[*] First-Order Deformations
[*] Problems
[/LIST]
[*] Further Reading
[/LIST]
[*] References
[*] Index of Notation
[/LIST]
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