Superfícies de Riemann e Curvas Algébricas

Objectives

Introduce the theory of Riemann Surfaces and Algebraic Curves as a foundation for the basic ideas of Algebraic Geometry over the complex numbers. The approach of the course is geometric.

Syllabus

Definition of a Riemann surface: complex charts and complex structures. Morphisms between Riemann surfaces. Basic examples. Hyperelliptic surfaces; gluing. Holomorphic and meromorphic functions on Riemann surfaces. Holomorphic maps between Riemann surfaces. Global properties of holomorphic maps. The Riemann-Hurwitz theorem. Holomorphic and meromorphic forms. Integration on Riemann surfaces. The Poincaré and Dolbeault lemmas. The residue theorem. Divisors. Linear equivalence of divisors. Plücker's formula. Spaces of functions and forms associated to a divisor. Algebraic curves. The Riemann- Roch theorem and the Serre duality. Applications. Presheaves, sheaves and Cech cohomology. Additional/optional topics: Singularities of algebraic curves. Classification of complex tori. Line bundles, the Picard group and the Jacobian variety. The theorems of Abel and Jacobi. Uniformization. Non-compact Riemann surfaces. Teichmüller space.

Prerequisites

Differential and Integral Calculus III and Introduction to Complex Analysis.

Cross Competence Component

The UC allows the development of transversal competences on Critical Thinking, Creativity and Problem Solving Strategies, in class, in autonomous work and in the several evaluation components. The percentage of the final grade associated with these competences should be around 15%.

Ethical Principles

All members of a group are responsible for the group’s work. In any assessment, every student shall honestly disclose any help received and sources used. In an oral assessment, every student shall be able to present and answer questions about the entire assignment and solution.