2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/78513I prove the existence, and describe the structure, of moduli space of pairs $(p,Θ)$ consisting of a projective variety $P$ with semiabelian group action and an ample Cartier divisor on it satisfying a few simple conditions. Every connected component of this moduli space is proper. A component containing a projective toric variety is described by a configuration of several polytopes, the main one of which is the secondary polytope. On the other hand, the component containing a principally polarized abelian variety provides a moduli compactification of $A_g$. The main irreducible component of this compactification is described by an "infinite periodic" analog of the secondary polytope and coincides with the toroidal compactification of $A_g$ for the second Voronoi decomposition.98 pages, published versionAlgebraic Geometry14K10text