For each universal genus‐g polarization μ of degree d, we construct a universal tropical Jacobian Jμ,gtrop as a generalized cone complex over the moduli space of stable pointed genus‐g tropical curves. We show several properties of the space Jμ,gtrop. In particular, we prove that the natural compactification of Jμ,gtrop is the tropicalization of the Esteves' compactified universal Jacobian over the moduli space of stable pointed genus‐g curves.

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In this paper we give local conditions to the existence of Abel maps for smoothings of nodal curves extending the Abel maps for the generic fiber. We use this result to construct Abel maps of any degree for nodal curves with two components. * The third author was partially supported by CNPq, processo 300714/2010-6.

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