Contents
Preface xi
Lecture 1. Rational and Polynomial Parametrizations 1
Lecture 2. Fractional Linear Transformations 11
Lecture 3. Cubic Curves 17
Lecture 4. Cubic Surfaces and General Hypersurfaces 23
Lecture 5. Outline of the Theory of Plane Curves 33
Lecture 6. Affine Plane and Projective Plane 41
Lecture 7. Sphere with Handles 47
Lecture 8. Functions and Differentials on a Curve 57
Lecture 9. Polynomials and Power Series 67
Lecture 10. Review of Abstract Algebra 75
Lecture 11. Some Commutative Algebra 83
Lecture 12. HenseFs Lemma and Newton's Theorem 89
Lecture 13. More about Newton's Theorem 95
Lecture 14. Branches and Valuations 99
Lecture 15. Divisors of Functions and Differentials 109
Lecture 16. Weierstrass Preparation Theorem 119
Lecture 17. Intersection Multiplicity 125
Lecture 18. Resolution of Singularities of Plane Curves 131
Lecture 19. Infinitely Near Singularities 145
Lecture 20. Parametrizing a Quartic with Three Double Points 159
Lecture 21. Characteristic Pairs 165
Lecture 22. Criterion For One Place and Jacobian Problem 177
Lecture 23. Inversion Formula and Jacobian Problem 189
Lecture 24. Surfaces 195
Lecture 25. Hypersurfaces 209
Lecture 26. Resolution of Singularities of Algebraic Surfaces 221
Lecture 27. Birational and Polyrational Transformations 235
Lecture 28. Valuations and Birational Correspondence 243
Lecture 29. Rational Cylinders through a Variety 255
Lecture 30. Resultants 267
Bibliography 275
Index 283
ix
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