Algebraic geometry II

Objectives

The course is an introduction to sheaves and schemes in algebraic geometry and their fundamental properties. This forms the basis of modern algebraic geometry.

Contents

The course gives an introduction to the theory of sheaves and schemes. In particular, the notions affine, noetherian, integral, reduced, irreducible, separated, proper and projective schemes are considered, as well as closed and open embeddings, sheaves of modules, divisors, morphisms into projective spaces, differentials, smooth schemes and Bertini's theorem.

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 sheaves and schemes, as well as morphisms between them
• is able to perform a simple analysis of schemes, in particular using properties of well-known sheaves.
• is able to produce the main ideas in the proofs of the most important results connected to the notions above.

Skills

The student

• masters fundamental techniques within sheaf and scheme theory, and morphisms between schemes, in particular embeddings of schemes into projective spaces.
• is able to argue mathematically correct and present proofs and reasoning
• has solid experience and training in reasoning with sheaves and geometric stuctures

General competence

The student

• is able to work individually and in groups
• is able to formulate in a precise and scientifically correct way
• is able to decide whether complex mathematical arguments are correct

Irregular, course will be offered if it is on this course list: Workbook: Emneliste for innreisende utvekslingsstudenter (uhad.no)