Tropicalizing vs Compactifying the Torelli morphism

TL;DR

This paper compares the compactified and tropical Torelli morphisms, analyzing their properties, fibers, and relationships through reduction and tropicalization maps to deepen understanding of their connections.

Contribution

It provides a detailed comparison of the compactified and tropical Torelli maps, clarifying their relationship and fiber structures through new reduction and tropicalization maps.

Findings

Descriptions of the fibers of both Torelli maps

Clarification of the relationship between the two maps

Introduction of reduction and tropicalization maps

Abstract

In this paper, we compare the compactified Torelli morphism (as defined by V. Alexeev) and the tropical Torelli map (as defined by the author in a joint work with S. Brannetti and M. Melo, and furthered studied by M. Chan). Our aim is twofold: on one hand, we will review the construction and main properties of the above mentioned two Torelli maps, focusing in particular on the description of their fibers achieved by the author in joint works with L. Caporaso; on the other hand, we will clarify the relationship between the two Torelli maps via the introduction of the reduction maps and the tropicalization maps.