Algebraic Geometry I

Objectives:

The course aims at giving an introduction to classical algebraic geometry, that is, the theory of algebraic varieties, and the most important notions and techinques to study these.

Contents:

The subject studies the Zariski topology and affine and projective varieties, as well as their regular functions, germs of regular functions, rational functions, dimension, Zariski tangent space and singularities. Moreover, also morphisms and rational maps between varieties and the differential of a morphism are studied, as well as simple intersection theory and Bezout's Theorem for curves in the projective plane. Some classical examples like products of varieties, blowing up of a point, the Segre embedding and the Veronese embedding are also presented.

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 results in algebraic geometry connected to the Zariski topology, affine and projective varieties, their regular functions, rational functions, Zariski tangent space and singularities, as well as morphisms and rational maps between varieties
• is able to perform an elementary analysis of simple varieties, in particular answer questions on irreducible components and singularieties
• knows fundamental intersection theory and Bezout's theorem
• has insight into important examples like products of varieties, blowing up of a point, the Segre embedding and the Veronese embedding
• is able to give an account of important connections bewteen geometry and commutative algebra
• 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 classical algebraic geometry
• is able to argue mathematically correct and present proofs and reasoning
• has solid experience and training in reasoning with 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)