The aim of this course is to study algebraic stacks and algebraic spaces, starting from a background in basic algebraic geometry [Hartshorne, Chapters II - III]. Although stacks were first introduced in order to solve moduli problems by recording automorphisms, they are now used throughout modern algebraic geometry and beyond! In particular, stacky techniques can often be employed to prove statements about schemes; for example a recent proof of resolution of singularities for algebraic varieties in characteristic zero uses stacks. Stacks also play an important role in (geometric) representation theory, such as in the Langlands program. Often moduli spaces and group actions are studied in relation with corresponding moduli stacks, particular from the perspective of enumerative geometry. There are well developed notions of coarse and good moduli spaces for stacks, which have lead to new constructions of moduli spaces when one does not have a natural set-up in terms of a group action. For a detailed list of the topics we plan to cover, please see the general information about the course.
- begleitende Lehrperson: Davide Gori
- verantwortliche Lehrperson: Victoria Hoskins