# Algebraic Curves and Riemann Surfaces by Miranda

• Geometry
• micromass
In summary, "Algebraic Curves and Riemann Surfaces" is a graduate level book that discusses the topic of algebraic curves and their relationship to Riemann surfaces. It requires knowledge in complex analysis, differential geometry, and abstract algebra. The book covers basic definitions of Riemann surfaces, examples of Riemann surfaces, projective curves, functions and maps on Riemann surfaces, group actions, integration, divisors and meromorphic functions, and the Riemann-Roch theorem. It also includes applications of Riemann surfaces in algebraic geometry and other fields.

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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]
[/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]
[/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]
[/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]
[/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]
[/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]
[/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]
[/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]
[/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]
[/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]
[/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]
[/LIST]
[*] References
[*] Index of Notation
[/LIST]

Last edited by a moderator:
One of the clearest books I know of to learn the topics in its title. I used this for my last course on Riemann surfaces and algebraic curves in 2010. I learned a lot myself and thoroughly enjoyed the reading. Good exercises too.

## 1. What are algebraic curves?

Algebraic curves are geometric objects studied in algebraic geometry that are defined by polynomial equations in two variables. They can be thought of as the set of points that satisfy a particular polynomial equation.

## 2. What are Riemann surfaces?

Riemann surfaces are one-dimensional complex manifolds, which are geometric objects that can be locally described by complex numbers. They are used in the study of complex analysis, algebraic geometry, and topology.

## 3. What is the connection between algebraic curves and Riemann surfaces?

There is a deep connection between algebraic curves and Riemann surfaces. In fact, every algebraic curve can be thought of as a Riemann surface, and vice versa. This connection allows for the use of tools and techniques from both algebraic geometry and complex analysis to study these objects.

## 4. Who is Miranda and what are their contributions to this topic?

Miranda is a mathematician who has made significant contributions to the study of algebraic curves and Riemann surfaces. Their book, "Algebraic Curves and Riemann Surfaces", is a well-known and widely used textbook on the subject. Miranda's work includes important results on the topology and classification of Riemann surfaces.

## 5. Is "Algebraic Curves and Riemann Surfaces" suitable for beginners?

The book "Algebraic Curves and Riemann Surfaces" is often used as a textbook for graduate level courses, so it may not be suitable for beginners. However, it does provide a thorough and comprehensive introduction to the topic, so with some prior knowledge of algebra and complex analysis, it could be accessible to beginners.

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