Geometric Invariant Theory is a central tool in the construction of moduli spaces, and shares the property ubiquitous among such tools that certain so-called 'unstable' objects must be excluded if the moduli space is to be well behaved. However, instability in GIT is a structured phenomenon: after making a choice of a certain invariant inner product, one has the HKKN stratification of the parameter space which, morally, sorts the objects according to how unstable they are. I will explain how one can use recent results of Berczi-Doran-Hawes-Kirwan in Non-Reductive GIT to perform quotients of these unstable strata as well, extending the classifications given by classical moduli spaces. This can be carried out, at least in principle, for any moduli problem that can be posed using GIT, and I will discuss two examples in particular: unstable (i.e. singular) curves, and coherent sheaves of fixed Harder-Narasimhan type. The latter of these is joint work with Gergely Berczi, Victoria Hoskins and Frances Kirwan.
- Algebraic Geometry Seminar