Given a tropical curve \(\Gamma\) of genus \(g\), one can canonically associate a tropical abelian variety to \(\Gamma\), namely its Jacobian \(\operatorname{Jac}(\Gamma)\). Since tropical curves form a moduli space \({M}_g^{\operatorname{trop}}\), the question for a universal tropical Jacobian, i.e. a generalized cone complex \(P\) with a natural map \(\pi : P \to {M}_g^{\operatorname{trop}}\) such that \(\pi^{-1}([\Gamma]) = \operatorname{Jac}(\Gamma) / \operatorname{Aut}(\Gamma)\), is natural. In the present paper the authors construct such a space, which can be thought of as a tropical analogue of Caporaso’s universal Picard scheme [L. Caporaso, J. Am. Math. Soc. 7, No. 3, 589–660 (1994; Zbl 0827.14014)].
The central tool developed in this article is the notion of \(\mu\)-polystable divisors. It is shown that, every divisor on a tropical curve \(\Gamma\) is linearly equivalent to a \(\mu\)-polystable divisor (Theorem 5.9) and two \(\mu\)-polystable divisors are linearly equivalent if and only if they are identified by an automorphism of \(\Gamma\) (Proposition 6.15). Since \(\mu\)-polystable divisors with fixed combinatorial type are parametrized by cones and specialization of combinatorial types induces face morphisms between these cones, the description of the universal Jacobian is straight forward in the end.
It is worth mentioning that this article is essentially a successor to previous work [A. Abreu and M. Pacini, Proc. Lond. Math. Soc. (3) 120, No. 3, 328–369 (2020; Zbl 1453.14082)] by a subset of the authors. The construction of a universal tropical Jacobian in the previous paper was based on the notion of \((p_0, \mu)\)-quasistability with respect to the choice of a base point \(p_0 \in \Gamma\). Hence the previous construction gave a universal Jacobian over the space of tropical curves with one marked point, whereas in the present paper, the result is defined over \(M_g^{\operatorname{trop}}\).