The Knop-Luna-Vust theory of spherical embeddings, extended to non-reductive groups

We extend the theory of spherical embeddings to actions of connected non-reductive groups. This generalization is formally very similar to the usual reductive case: equivariant embeddings are described essentially by collections of convex cones in a rational vector space.
We also show some new relationship between the combinatorics emerging in this case and the properties of the unipotent radical of the acting group. Finally, we apply our techniques to prove a characterization of log homogeneous varieties.