Algebraic Geometry

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.

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.

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

The course will be taught as needed. If there are few PhD students, the course is only given as a guided self-study.