We explore different projective moduli spaces of semistable hypersurfaces in toric varieties for differing stability conditions.
In particular we look at K-stability and (Non-reductive) Geometric Invariant Theory, in the latter we translate the problem to one of polytopes. In a few cases of hypersurfaces in weighted projective space we compute the cohomology of the corresponding moduli space.
This is based on joint work with Joshua Jackson.