Fibrations represented by double divisors on the base space

In two different situations we describe fibrations via divisors on the base space with rather unusual coefficients: First, branched coverings X -> P1 that are the quotient of the action of a finite group A can be encoded by a degree zero divisor on P1 with coefficients in A.
Second, degenerate toric fibrations X -> Y with generic fiber F = TV(sigma) (the toric variety associated to a cone sigma) correspond to a divisor on Y with coefficients being polyhedra in the space N of one-parameter subgroups of the torus.
In the latter case, the polyhedral coefficients can alternatively understood as divisors on modifications of the toric variety TV(sigma-dual). This is refered to by the name "double divisors" in the title.