This book introduces the contemporary notions of algebraic varieties, morphisms of varieties, and adeles to the classical subject of plane curves over algebraically closed fields. It is useful for advanced undergraduate and beginning graduate students in mathematics.
1. Prerequisites
2. Some Facts About Polynomials
3. Affine Plane Curves
4. Tangent Spaces
5. The Local Ring at a Point
6. Projective Plane Curves
7. Rational Mappings, Birational Correspondences and Isomorphisms of Curves
8. Examples of Rational Curves
9. The Correspondence between Valuations and Points
10. An Overview and Sideways Glance
11. Divisors
12. The Divisor of a Function Has Degree
13. Riemann's Theorem
14. The Genus of a Nonsingular Plane Curve
15. Curves of Genus 0 and 1
16. A Classification of Isomorphism Classes of Curves of Genus 1
17. The Genus of a Singular Curve
18. Inflection Points on Plane Curves
19. Bezout's Theorem
20. Addition on a Nonsingular Cubic
21. Derivations, Differentials and the Canonical Class
22. Adeles and the Riemann-Roch Theorem
