Valery Alexeev - Families of algebraic varieties associated with cell decompositions

Fix a finite set S of integral points in a lattice L and let Q be their convex hull. To any cell decomposition of Q with vertices in S we associate: 1)  a family of projective algebraic varieties, and 2)  a family of pairs of projective algebraic varieties together with Cartier divisors.

Similarly, we construct families of the above two types for any periodic cell decomposition with integral vertices.

The parameter spaces of families of the second type for various cell decompositions fit together into a moduli space, a proper algebraic variety itself. One application of this construction is a canonical geometric compactification of the moduli of principally polarized abelian varieties.