Valuative criteria for stability in unstable Θ-Strata

In the first part of this thesis, we give a criterion for general Θ-strata to admit moduli spaces. This makes it possible to construct moduli spaces for unstable strata of a stack that doesn’t come with a quotient presentation. We use this criterion to deduce a version of the ˆU -theorem on the existence of geometric quotients for actions of a graded unipotent group over a Noetherian base scheme. In the second part, we show that an algebraic group with a graded unipotent radical admits decompositions similar to Cartan decompositions of reductive groups. We give a geometric interpretation of these decompositions in terms of a stacky valuative criterion. Using this geometric interpretation we show that the quotient stack of an algebraic group with graded unipotent radical acting on a projective scheme is almost proper.