Course content

The course introduces the central concepts of algebraic geometry. Affine and projective varieties are introduced, and these and their morphisms are studied. The concept of a sheaf on a topological space is introduced, and in particular affine and projective varieties are interpreted as the locally ringed spaces. Sheaves of modules and the concept of sheafification are discussed. Beyond these basic concepts the content of the course may vary, and include for instance divisors, resolutions of singularities, Riemann-Roch theorem for curves, elliptic curves, Bezout’s theorem, sheaf cohomology, schemes.

Learning outcome

1. Knowledge. The student knows the basic concepts of algebraic geometry, in particular algebraic varieties with their structure sheaves, and the categories of coherent sheaves on these. Further the student is familiar with some more advanced subjects, depending on the course content that year. 2. Skills. The students should learn the topics mentioned above and be able to apply these concepts in own research.

Learning methods and activities

Lectures, some tasks for homework, alternatively as guided self-study.

The course is taught every third year, next time Spring 2027. If there are few students, there will be guided self-study.

Compulsory assignments

• Exercises

Recommended previous knowledge

MA8202 Commutative algebra.

Required previous knowledge

Participants need some knowledge of (commutative) rings and modules, in particular the definitions of these. Moreover they should know the concept of localization of a commutative ring with respect to a multiplicative subset.

Course materials

Will be announced at the start of the course.