Non-reductive geometric invariant theory and compactifications of enveloped quotients

In this thesis we develop a framework for constructing quotients of varieties by actions of linear algebraic groups which is similar in spirit to that of Mumford's geometric invariant theory. This is done by extending the work of Doran and Kirwan in the unipotent setting to deal with more general non-reductive groups. Given a linear algebraic group acting on an irreducible variety with a linearisation, an open subset of stable points is identified that admits a geometric quotient in the category of varieties. This lies within the enveloped quotient, which is a dense constructible subset of a scheme that is locally of finite type, called the enveloping quotient. Ways to compactify the enveloped quotient---and the quotient of the stable locus therein---are considered. In particular, the theory of reductive envelopes from Doran and Kirwan's work is extended to the more general non-reductive setting to give ways of constructing compactifications of the enveloped quotient by using the techniques of Mumford's geometric invariant theory for reductive groups. We then look at two ways in which this non-reductive geometric invariant theory can be used in practice. Firstly, we consider a procedure for constructing quotients inductively, using the extra data of a choice of appropriate subnormal series of a group. Related to this is a method for constructing an approximation of the stable set. Secondly, we study the actions of certain extensions of unipotent groups by multiplicative groups on projective varieties with very ample linearisation. Here we identify an open subset of points that admits a geometric quotient by the action of the extended group and which is explicitly computable via Hilbert-Mumford-like criteria.