NMAK22018U Algebraic Geometry (AlgGeo)

Volume 2025/2026
Education

MSc Programme in Mathematics

MSc Programme in Mathematics with a minor subject

Content

Algebraic Geometry is the study of geometric structures arising from solution sets of polynomial equations, and forms a central part of modern mathematics. It has numerous applications, ranging from number theory to theoretical physics.

The course will be an introduction to Algebraic Geometry, and will cover the following topics:
Algebraic sets, affine and projective varieties, fundamental properties of varieties. Sheaves and locally ringed spaces. Morphisms of varieties, birational maps and blow-ups. Smoothness and singularities. Hilbert polynomials and Bezout's theorem.

Learning Outcome

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

Knowledge of advanced linear algebra (e.g., obtained through the UCPH MSc course Advanced vector spaces (AdVec)).

Knowledge about general topology (e.g., obtained through UCPH BSc course Topology) and commutative algebra (e.g., obtained through UCPH MSc course Commutative Algebra).

Academic qualifications equivalent to a BSc degree is recommended.
4 hours lectures and 4 hours exercises per week for 9 weeks
  • Category
  • Hours
  • Lectures
  • 36
  • Preparation
  • 134
  • Exercises
  • 36
  • Total
  • 206
Written
Oral
Individual
Collective
Continuous feedback during the course of the semester
Credit
7,5 ECTS
Type of assessment
Continuous assessment
Type of assessment details
Seven weekly homework assignments of which the five highest scores count towards the final grade, weighted equally at 10% each, and a three-hour closed-book final in-class problem set, accounting for 50% of the grade
Aid
Only certain aids allowed (see description below)

Weekly homework: 
All aids except generative artificial intelligence are allowed for the weekly homework.

 

Three-hour final in-class problem set:
No books, notes, or electronic aids are allowed for the three-hour final in-class problem set, apart from one personally created handwritten one-sided A4 page of notes.

Marking scale
7-point grading scale
Censorship form
No external censorship
One internal examiner
Re-exam

30 minutes oral examination with no aids or preparation time. The oral examination will cover the entire material of the course, including the exercise sets.

Criteria for exam assesment

The student should convincingly and accurately demonstrate the knowledge, skills and competences described under Intended learning outcome.