复代数曲面

　　1 Introductionandbackground
　　1.1 Abriefhistoryofalgebraiccurves
　　1.2 Relationshipwithotherpartsofmathematics
　　1.2.1 Numbertheory
　　1.2.2 Singularitiesandthetheoryofknots
　　1.2.3 Complexanalysis
　　1.2.4 Abelianintegrals
　　1.3 RealAlgebraicCurves
　　1.3.1 Hilbert'sNullstellensatz
　　1.3.2 Techniquesfordrawingrealalgebraiccurves
　　1.3.3 Realalgebraiccurvesinsidecomplexalgebraiccurves
　　1.3.4 Importantexamplesofrealalgebraiccurves
　　2 Foundations
　　2.1 ComplexalgebraiccurvesinCs
　　2.2 Complexprojectivespaces
　　2.3 ComplexprojectivecurvesinPs
　　2.4 Affineandprojectivecurves
　　2.5 Exercises
　　3 Algebraicproperties
　　3.1 Bezout'stheorem
　　3.2 Pointsofinflectionandcubiccurves
　　3.3 Exercises
　　4 Topologicalproperties
　　4.1 Thedegree-genusformula
　　4.1.1 Thefirstmethodofproof
　　4.1.2 Thesecondmethodofproof
　　4.2 BranchedcoversofPI
　　4.3 Proofofthedegree-genusformula
　　4.4 Exercises
　　5 Riemannsurfaces
　　5.1 TheWeierstrassfunction
　　5.2 Riemannsurfaces
　　5.3 Exercises
　　6 DifferentialsonRiemannsurfaces
　　6.1 Holomorphicdifferentials
　　6.2 Abel'stheorem
　　6.3 TheRiemann-Rochtheorem
　　6.4 Exercises
　　7 Singularcurves
　　7.1 ResolutionofSingularities
　　7.2 NewtonpolygonsandPuiseuxexpansions
　　7.3 Thetopologyofsingularcurves
　　7.4 Exercises
　　AAlgebra
　　BComplexanalysis
　　CTopology
　　C.1Coveringprojections
　　C.2Thegenusisatopologicalinvariant
　　C.3Sphereswithhandles

　　    This book on complex algebraic curves is intended to be accessible to any third year mathematics undergraduate who has attended courses on algebra，topology and complex analysis. It is an expanded version of notes written to accompany a lecture course given to third year undergraduates at Oxford.It has usually been the case that a number of graduate students have also attended the course， and the lecture notes have been extended somewhat for the sake of others in their position. However this new material is not intended to daunt undergraduates， who can safely ignore it. The original lecture course consisted of Chapters 1 to 5 （except for some of §3.1 including the definition of intersection multiplicities） and part of Chapter 6， although some of the contents of these chapters （particularly the introductory material in Chapter 1） was covered rather briefly.