algebraic geometry III

Objectives

The course is an introduction to the cohomology of sheaves, as well as algebraic curves and surfaces within algebraic geometry.

Contents

The course gives an introduction to the theory of cohomology of sheaves applied to schemes, like for instance Cech cohomology and Serre duality, as well as basic theory of algebraic curves, such as the Riemann-Roch theorem, the Riemann-Hurwitz theorem, Clifford’s theorem, embeddings into projective spaces, and of algebraic surfaces, like intersection theory, the Riemann-Roch theorem and blowing ups.

On completion of the course the student should have the following learning outcomes defined in terms of knowledge, skills and general competence:

Knowledge

The student

• is able to define and use fundamental notions and constructions and knows important results in algebraic geometry connected to the cohomology of sheaves, algebraic curves and surfaces, as well as morphisms between them
• is able to analyse schemes and morphisms using cohomology of sheaves
• is able to produce the main ideas in the proofs of the most important results connected to the notions above.

Skills

The student

• is able to use the fundamental techniques that are important in many problems in algebraic geometry
• is able to produce short proofs of statements in algebraic geometry
• has solid experience and training in reasoning with abstract mathematical stuctures

General competence

The student

• is able to read a research article in algebraic geometry independently (with some effort).
• is able to follow the introduction of a research talk in algebraic geometry.
• is able to follow a colloquium talk in algebraic geometry.