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Geometry Algebraic Curves and Riemann Surfaces by Miranda

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  1. Feb 1, 2013 #1

    micromass

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    Table of Contents:
    Code (Text):

    [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]
     
     
    Last edited by a moderator: May 6, 2017
  2. jcsd
  3. Feb 2, 2013 #2

    mathwonk

    User Avatar
    Science Advisor
    Homework Helper

    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.
     
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