Vector bundles on algebraic curves
This course will be a joint course together with G. Hein.
Time and Place:
Tuesday 14-16 WSC-S-U-4.01
Friday 14-16 WSC-S-U-3.03
depending on the participants we might add an exercise session.
There is a moodle page for this course. (Vectorbundles)
Prerequisites: This course is aimed towards students in Algebraic Geometry. Participants should have completed a basic course in Algebraic Geometry or Complex Geometry. The course can serve as a good starting point for a master thesis.
Content: A very rich part of algebraic geometry comes from the observation that objects can often be studied in families and often even the collection of all objects of some given type comes equipped with a natural structure of an algebraic variety, which is the called "moduli space". The most basic example of this is projective space, which parametrizes lines in a vector space.
As an excuse to learn about various methods to study moduli spaces we will consider vector bundles on curves. This provides a very rich example that has been studied form an extremely wides range of perspectives and reasons. It has the advantage to be on the one hand acessible to detailed analysis, for example a large chunck of the problem can be described in terms of linear algebra, and on the other hand the problem contains many interesting properties, e.g. it is not so easy to parametrize all objects at the same time and special objects have a large amount of symmetry which causes peculiar geometric structure of the moduli space around those points.
In the beginning we will cover and review basic results on curves, sheaves, vector bundles, to accomodate different kinds of background of the participants.