Ⅰ History of algebraic curves 1. Origin and generation of curves 1.1 The circle and the straight line 1.2 The classical problems of antiquity 1.3 The conic sections 1.4 The cissoid of Diocles 1.5 The conchoid of Nicomedes 1.6 The spiric sections of Perseus 1.7 From the epicycles of Hipparchos to the Wankel motor 1.8 Caustics and contour curves in optics and perspective 1.9 Further examples of curves from science and technology 2. Synthetic and analytic geometry 2.1 Coordinates 2.2 The development of analytic geometry 2.3 Equations for curves 2.4 Examples of the application of analytic methods 2.5 Newton's investigation of cubic curves 3. The development of projective geometry 3.1 Descriptive geometry and projective geometry 3.2 The development of analytic projective geometry 3.3 The projective plane as a manifold 3.4 Complex projective geometry Ⅱ Investigation of curves by elementary algebraic methods 4. Polynomials 4.1 Decomposition into prime factors 4.2 Divisibility properties of polynomials 4.3 Zeroes of polynomials 4.4 Homogeneous and inhomogeneous polynomials 5. Definition and elementary properties of plane algebraic curves 5.1 Decomposition into irreducible components 5.2 Intersection of a curve by a line 5.3 Singular points of plane curves 6. The intersection of plane curves 6.1 Bezout's theorem 6.2 Applications of Bezout's theorem 6.3 The intersection ring of P2(c) 7. Some simple types of curves 7.1 Quadrics. 7.2 Linear systems of cubics 7.3 Inflection point figures and normal forms of cubics 7.4 Cubics, elliptic curves and abelian varieties Ⅲ Investigation of curves by resolution of singularities 8. Local investigations 8.1 Localisation-local rings 8.2 Singularities as analytic set germs 8.3 Newton polygons and Puiseux expansions 8.4 Resolution of singularities by quadratic transformations 8.5 Topology of singularities 9. Global investigations 9.1 The Plucker formulae
9.2 The formulae of Clebsch and Noether 9.3 Differential forms on Riemann surfaces and their periods Bibliography Index
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