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中譯名： 復代數麯麵

世圖書號： 978-7-5062-9203-0

原版書號： 978-0-521-42353-3

原齣版社： Cambridge University Press

原版齣版年代： 1992年

世圖影印年代： 2008年

目錄及部分內容頁要覽：

19世紀發展起來的復代數麯麵理論，其良好的性質已經在數學的各個領域以及理論物理學中得到很好的應用，成為許多科目研究中心話題。本書源自Kirwan 在牛津大學的講義，作者以本科生掌握的數學知識為基礎引入瞭該理論，詳細介紹瞭復代數麯麵的代數和拓撲性質以及它們和復分析的聯係。本書適於數學專業本科高年級研究生以及相關專業的研究人員。

目次：背景；基礎知識；代數性質；拓撲性質；黎曼麵；黎曼麵上的微分；奇異麯麵。

Dame Frances Clare Kirwan, DBE FRS (born 1959) is a British mathematician, currently a Professor of Mathematics at the University of Oxford. Her fields of specialisation are algebraic and symplectic geometry.

http://en.wikipedia.org/wiki/Frances_Kirwan

https://www.maths.ox.ac.uk/people/profiles/frances.kirwan

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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