Complex Algebraic Curves

Complex Algebraic Curves
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作者:
1992-05
ISBN: 9780521423533
装帧: 其他
开本: 其他
纸张: 其他
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  • Complex algebraic curves were developed in the nineteenth century. They have many fascinating properties and crop up in various areas of mathematics, from number theory to theoretical physics, and are the subject of much research. By using only the basic  1 Introduction and background
    1.1 A brief history of algebraic curves
    1.2 Relationship with other parts of mathematics
    1.2.1 Number theory
    1.2.2 Singularities and the theory of knots
    1.2.3 Complex analysis
    1.2.4 Abelian integrals
    1.3 Real Algebraic Curves
    1.3.1 Hilbert's Nullstellensatz
    1.3.2 Techniques for drawing real algebraic curves
    1.3.3 Real algebraic curves inside complex algebraic curves
    1.3.4 Important examples of real algebraic curves
    2 Foundations
    2.1 Complex algebraic curves in Cs
    2.2 Complex projective spaces
    2.3 Complex projective curves in Ps
    2.4 Affine and projective curves
    2.5 Exercises
    3 Algebraic properties
    3.1 Bezout's theorem
    3.2 Points of inflection and cubic curves
    3.3 Exercises
    4 Topological properties
    4.1 The degree-genus formula
    4.1.1 The first method of proof
    4.1.2 The second method of proof
    4.2 Branched covers of PI
    4.3 Proof of the degree-genus formula
    4.4 Exercises
    5 Riemann surfaces
    5.1 The Weierstrass function
    5.2 Riemann surfaces
    5.3 Exercises
    6 Differentials on Riemann surfaces
    6.1 Holomorphic differentials
    6.2 Abel's theorem
    6.3 The Riemann-Roch theorem
    6.4 Exercises
    7 Singular curves
    7.1 Resolution of Singularities
    7.2 Newton polygons and Puiseux expansions
    7.3 The topology of singular curves
    7.4 Exercises
    A Algebra
    B Complex analysis
    C Topology
    C.1 Covering projections
    C.2 The genus is a topological invariant
    C.3 Spheres with handles
  • 内容简介:
    Complex algebraic curves were developed in the nineteenth century. They have many fascinating properties and crop up in various areas of mathematics, from number theory to theoretical physics, and are the subject of much research. By using only the basic 
  • 目录:
    1 Introduction and background
    1.1 A brief history of algebraic curves
    1.2 Relationship with other parts of mathematics
    1.2.1 Number theory
    1.2.2 Singularities and the theory of knots
    1.2.3 Complex analysis
    1.2.4 Abelian integrals
    1.3 Real Algebraic Curves
    1.3.1 Hilbert's Nullstellensatz
    1.3.2 Techniques for drawing real algebraic curves
    1.3.3 Real algebraic curves inside complex algebraic curves
    1.3.4 Important examples of real algebraic curves
    2 Foundations
    2.1 Complex algebraic curves in Cs
    2.2 Complex projective spaces
    2.3 Complex projective curves in Ps
    2.4 Affine and projective curves
    2.5 Exercises
    3 Algebraic properties
    3.1 Bezout's theorem
    3.2 Points of inflection and cubic curves
    3.3 Exercises
    4 Topological properties
    4.1 The degree-genus formula
    4.1.1 The first method of proof
    4.1.2 The second method of proof
    4.2 Branched covers of PI
    4.3 Proof of the degree-genus formula
    4.4 Exercises
    5 Riemann surfaces
    5.1 The Weierstrass function
    5.2 Riemann surfaces
    5.3 Exercises
    6 Differentials on Riemann surfaces
    6.1 Holomorphic differentials
    6.2 Abel's theorem
    6.3 The Riemann-Roch theorem
    6.4 Exercises
    7 Singular curves
    7.1 Resolution of Singularities
    7.2 Newton polygons and Puiseux expansions
    7.3 The topology of singular curves
    7.4 Exercises
    A Algebra
    B Complex analysis
    C Topology
    C.1 Covering projections
    C.2 The genus is a topological invariant
    C.3 Spheres with handles
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