  We describe limits of line bundles on nodal curves in terms of toric
arrangements associated to Voronoi tilings of Euclidean spaces. These tilings
encode information on the relationship between the possibly infinitely many
limits, and ultimately give rise to a new definition of limit linear series.
This article and the first two that preceded it are the first in a series aimed
to explore this new approach.
  In Part I, we set up the combinatorial framework and showed how graphs
weighted with integer lengths associated to the edges provide tilings of
Euclidean spaces by certain polytopes associated to the graph itself and to its
subgraphs.
  In Part II, we described the arrangements of toric varieties associated to
the tilings of Part I in several ways: using normal fans, as unions of orbits,
by equations and as degenerations of tori.
  In the present Part III, we show how these combinatorial and toric frameworks
allow us to describe all stable limits of a family of line bundles along a
degenerating family of curves. Our main result asserts that the collection of
all these limits is parametrized by a connected 0-dimensional closed substack
of the Artin stack of all torsion-free rank-one sheaves on the limit curve.
Moreover, we thoroughly describe this closed substack and all the closed
substacks that arise in this way as certain torus quotients of the arrangements
of toric varieties of Part II determined by the Voronoi tilings of Euclidean
spaces studied in Part I.
