复代数曲面

1Introductionandbackground
1.1Abriefhistoryofalgebraiccurves
1.2Relationshipwithotherpartsofmathematics
1.2.1Numbertheory
1.2.2Singularitiesandthetheoryofknots
1.2.3Complexanalysis
1.2.4Abelianintegrals
1.3RealAlgebraicCurves
1.3.1Hilbert'sNullstellensatz
1.3.2Techniquesfordrawingrealalgebraiccurves
1.3.3Realalgebraiccurvesinsidecomplexalgebraiccurves
1.3.4Importantexamplesofrealalgebraiccurves

2Foundations
2.1ComplexalgebraiccurvesinCs
2.2Complexprojectivespaces
2.3ComplexprojectivecurvesinPs
2.4Affineandprojectivecurves
2.5Exercises

3Algebraicproperties
3.1Bezout'stheorem
3.2Pointsofinflectionandcubiccurves
3.3Exercises

4Topologicalproperties
4.1Thedegree-genusformula
4.1.1Thefirstmethodofproof
4.1.2Thesecondmethodofproof
4.2BranchedcoversofPI
4.3Proofofthedegree-genusformula
4.4Exercises

5Riemannsurfaces
5.1TheWeierstrassfunction
5.2Riemannsurfaces
5.3Exercises

6DifferentialsonRiemannsurfaces
6.1Holomorphicdifferentials
6.2Abel'stheorem
6.3TheRiemann-Rochtheorem
6.4Exercises

7Singularcurves
7.1ResolutionofSingularities
7.2NewtonpolygonsandPuiseuxexpansions
7.3Thetopologyofsingularcurves
7.4Exercises

AAlgebra
BComplexanalysis
CTopology
C.1Coveringprojections
C.2Thegenusisatopologicalinvariant
C.3Sphereswithhandles