This course studies moduli problems in algebraic geometry and the construction of moduli spaces as group quotients using geometric invariant theory (GIT) as developed by Mumford. Moduli theory involves classifying certain objects up to natural equivalences or isomorphism, and ideally one wants a scheme whose points parametrise the equivalence classes and whose geometry describes how objects vary and deform. After some motivation from moduli theory, we will study group actions in algebraic geometry and GIT quotients, which are locally on open affine sets given by taking invariant rings. In particular, we will study Hilbert's 14th problem of when invariant rings are finitely generated, which will lead to a digression on reductive and non-reductive groups. After describing quotients of affine schemes, we move to a more general setting, which involves choosing an equivariant line bundle (or equivariant projective embedding) to find an open semistable locus that can be covered by open affine invariant subsets whose quotients we can glue. A crucial result in the course is the Hilbert-Mumford criterion, which gives a combinatorial way to interpret the semistable locus in certain settings. Throughout the course, we will study many different examples, and towards the end of the course, we will apply these techniques to construct moduli spaces of projective hypersurfaces and moduli spaces of semistable vector bundles on a smooth projective curve.
- verantwortliche Lehrperson: Victoria Hoskins