A universal tropical Jacobian over Mgtrop$M_g^{\rm {trop}}$

Abstract

We introduce and study polystable divisors on a tropical curve, which are the tropical analogue of polystable torsion‐free rank‐1 sheaves on a nodal curve. We construct a universal tropical Jacobian over the moduli space of tropical curves of genus g$g$. This space parametrizes equivalence classes of tropical curves of genus g$g$ together with a μ$\mu$‐polystable divisor, and can be seen as a tropical counterpart of Caporaso universal Picard scheme. We describe polyhedral decompositions of the Jacobian of a tropical curve via polystable divisors, relating them with other known polyhedral decompositions.

The moduli space of quasistable spin curves

“…By [AP20, Theorem 5.10] we have that J(X) and J qs v0 (X) are homeomorphic. The structure of a polyhedral complex for the tropical Jacobian was first described in [ABKS14], and was extended in [AP20], [CPS23] and [AAPT23].…”