Kursussøgning, efter- og videreuddannelse – Københavns Universitet

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Kursussøgning, efter- og videreuddannelse

CANCELLED Rational Surfaces

Practical information
Study year 2016/2017
Block 2
Programme level Full Degree Master
Course responsible
  • Sho Tanimoto (3-83787f507d7184783e7b853e747b)
  • Department of Mathematical Sciences
Course number: NMAK16016U

Course content

Roughly speaking, rational surfaces are surfaces which can be parametrized by two parameters. Their geometry is very explicit, but it is also very rich, so these surfaces provide very nice examples which we can apply many techniques from  algebraic geometry. Also one can consider many arithmetic problems regarding rational surfaces.

This course will be an introduction to theory of rational surfaces. It will cover the following topics:

Cohomology of line bundles, intersection theory on surfaces, blow ups and the contractibility criterion, del Pezzo surfaces,  Castelnuovo’s theorem, the minimal model program in dimension 2, classification of minimal rational surfaces over a non-closed field, rational points over function fields.

Learning outcome

Knowledge: To display knowledge and understanding of the course topics 
and content at a level suitable for further studies in algebraic and arithmetic geometry

Skills: At the end of the course the student is expected to be able to 
follow and reproduce arguments at a high abstract level corresponding to 
the contents of the course. 

Competences: At the end of the course the student is expected to be 
able to apply basic techniques and results to concrete examples.

Recommended prerequisites

Knowledge about Algebraic Geometry

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MSc Programme in Mathematics


Study Board of Mathematics and Computer Science

Course type

Single subject courses (day)


1 block


---- SKEMA LINK ----

Teaching and learning methods

5 hours lectures and 3 hours exercises each week for 7 weeks


No restrictions/ no limitations




Category Hours
Lectures 35
Exercises 21
Exam 1
Preparation 149
English 206


Type of assessment

Oral examination under invigilation
Oral examination, 30 min
The student will have 30 minutes preparation before the exam.


All aids allowed

Marking scale

7-point grading scale

Criteria for exam assessment

The student must in a satisfactory way demonstrate that he/she has mastered the learning outcome of the course.

Censorship form

No external censorship
Several internal examiners.


The same as the ordinary exam. 
To be eligible to take the re-exam, students who have not already had the mandatory assignment approved must re-submit the assignment no later than 2 weeks before the re-exam week. The mandatory assignment must be approved in order to take the re-exam.

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