Abel maps for nodal curves via tropical geometry

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. Finally, we write an algorithm, which we implemented in SageMath to compute explicitly the solution of the combinatorial problem which, provided the existence of certain subdivisions of a hypercube, give rise to the resolution of the geometric Abel map.