Moduli spaces of unstable curves and sheaves via non-reductive geometric invariant theory

Many moduli problems in algebraic geometry can be posed using Geometric Invariant Theory (GIT). However, as with all such tools, if we are to have any hope of obtaining a well-behaved moduli space, certain objects, which are in whatever sense degenerate, must be ruled 'unstable' and left out of the classification. The goal of this thesis is to show how new results in non-reductive GIT may be used to construct moduli spaces that parameterise these unstable objects. Once a moduli problem is posed using GIT, this is done by studying the associated instability stratification and using non-reductive GIT to take quotients of the unstable strata, effectively splitting the moduli problem up into manageable pieces. We apply this method in two examples. First, we consider moduli spaces of coherent sheaves over a projective scheme. The instability stratification in this case is closely related to the stratification by Harder-Narasimhan type. However, in order to apply the relevant theorems in non-reductive GIT, in fact one must perform a blow-up process analogous to partial desingularisation in the reductive case. This corresponds to refining the stratification by Harder-Narasimhan type. We give a sheaf-theoretic interpretation of this refinement, and use it to construct moduli spaces of such sheaves, in the case of Harder-Narasimhan length 2. We also include some results towards the length 3 case, and some remarks on generalisation to arbitrary lengh. Our second application is moduli of projective algebraic curves. Here it is not clear a priori what the instability stratification should mean intrinsically, which leads us to define the notion of Rosenlicht-Serre type for curves. We show that in the case of a curve with only unibranch singularities this data essentially coincides with the semigroup of the singularities, and give a construction of moduli spaces of unibranch curves of fixed Rosenlicht-Serre type.