Geometric invariant theory and stacks

Abstract

A moduli problem can often be translated to the problem of constructing a certain type of quotient. Mumford's geometric invariant theory uses ideas from classical invariant theory to construct such quotients. In particular, Mumford shows that for the action of a reductive group on a scheme X we can construct quotients for the subsets of so called (semi-)stable points of X. Moreover, he provides a numerical criterion for identifying these (semi-)stable points. Many decades after geometric invariant theory was first introduced, Alper introduced the notion of a good moduli space and gave a generalization of Mumford’s GIT to the setting of algebraic stacks. In this thesis, we will discuss all the above. Furthermore, we will dedicate a chapter to an existence result for good moduli spaces by Alper, Halpern-Leistner and Heinloth, and another chapter to Heinloth's reformulation of the numerical criterion for determining stability in terms of algebraic stacks.