Algebraic Curves and Riemann Surfaces by Miranda

[micromass]

• 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:

[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]

 

[mathwonk]

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.