  We consider Abel maps for regular smoothing of nodal curves with values in
the Esteves compactified Jacobian. In general, these maps are just rational,
and an interesting question is to find an explicit resolution. We translate
this problem into an explicit combinatorial problem by means of tropical and
toric geometry. We show that the solution of the combinatorial problem gives
rise to an explicit resolution of the Abel map. We are able to use this
technique to construct and study all the Abel maps of degree one.
