  The Losev-Manin moduli space parametrizes pointed chains of projective lines.
In this paper we study a possible generalization to families of pointed
degenerate toric varieties. Geometric properties of these families, such as
flatness and reducedness of the fibers, are explored via a combinatorial
characterization. We show that such families are described by a specific type
of polytope fibration which generalizes the twisted Cayley sums, originally
introduced to characterize elementary extremal contractions of fiber type
associated to projective $\mathbb{Q}$-factorial toric varieties with positive
dual defect. The case of a one-dimensional simplex can be viewed as an
alternative construction of the permutohedra.
