Tropicalization of the universal Jacobian

TL;DR

This paper develops a stack-theoretic framework to understand the tropicalization of the universal Jacobian over tropical curves, establishing compatibility between logarithmic and non-Archimedean approaches.

Contribution

It introduces two approaches to tropicalization of the universal compactified Jacobian and proves their compatibility, advancing the understanding of tropical Jacobians in algebraic geometry.

Findings

Tropicalization of the universal compactified Jacobian equals the universal tropical Jacobian.

The tropicalization maps are compatible with tautological morphisms.

Framework sets the stage for explicit polyhedral models of the logarithmic Picard variety.

Abstract

In this article we provide a stack-theoretic framework to study the universal tropical Jacobian over the moduli space of tropical curves. We develop two approaches to the process of tropicalization of the universal compactified Jacobian over the moduli space of curves -- one from a logarithmic and the other from a non-Archimedean analytic point of view. The central result from both points of view is that the tropicalization of the universal compactified Jacobian is the universal tropical Jacobian and that the tropicalization maps in each of the two contexts are compatible with the tautological morphisms. In a sequel we will use the techniques developed here to provide explicit polyhedral models for the logarithmic Picard variety.