In this course we are going to study algebraic curves, the most ancient algebraic objects. The tools used in algebraic geometry are very abstract. So we are going to see how these tools apply in the case of algebraic curves.

Program:

1. Algebraic curves. Their description. Definitions: Riemannian surfaces, rational and regular functions, ring and field of functions.
2. Divisors. Linear bundles. Divisors on curves.
3. Ramifications. The Riemann-Hurwitz formula.
4. Canonical class. The Riemann-Roch theorem.
5. Bijection between non-singular algebraic curves and field extensions of transcendence degree 1.
6. Elliptic curves. The elliptic curve group law.
7. Automorphisms of algebraic curves. The Hurwitz theorem. The Macbeath theorem.
8. Jacobians of curves.
9. The Torelli theorem.

Lecturer
Vikulova Anastasia Vadimovna

Organizations
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
Steklov International Mathematical Center