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Postgraduate course

Algebraic Geometry I

  • ECTS credits10
  • Teaching semesterSpring, Autumn
  • Course codeMAT229
  • Number of semesters1
  • Language

    English if English-speaking students attend the seminars, otherwise Norwegian

  • Resources

Teaching semester

Irregular

Objectives and Content

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.

Learning Outcomes

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

Recommended Previous Knowledge

MAT224 and one of MAT242 or MAT243

Compulsory Assignments and Attendance

Excercises

Forms of Assessment

Oral examination

Grading Scale

The grading scale used is A to F. Grade A is the highest passing grade in the grading scale, grade F is a fail.

Subject Overlap

MAT321: 10 ECTS

Exam information

  • Type of assessment: Oral examination

    Withdrawal deadline
    01.11.2018