The universal tropical Jacobian and the skeleton of the Esteves' universal Jacobian

TL;DR

This paper constructs a universal tropical Jacobian for each polarization of a fixed genus and degree, demonstrating its properties and its relation to the tropicalization of Esteves' universal Jacobian.

Contribution

It introduces a universal tropical Jacobian as a generalized cone complex and proves its compactification matches the tropicalization of Esteves' universal Jacobian.

Findings

Construction of the universal tropical Jacobian as a generalized cone complex

Proof that the compactification is the tropicalization of Esteves' Jacobian

Properties of the tropical Jacobian space

Abstract

For each universal genus-$g$ polarization $\mu$ of degree $d$, we construct a universal tropical Jacobian $J_{\mu,g}^{trop}$ as a generalized cone complex over the moduli space of stable pointed genus-$g$ tropical curves. We show several properties of the space $J_{\mu,g}^{trop}$. In particular, we prove that the natural compactification of $J_{\mu,g}^{trop}$ is the tropicalization of the Esteves' compactified universal Jacobian over the moduli space of stable pointed genus-$g$ curves.