Faithful Tropicalization of Mumford Curves of Genus Two

TL;DR

This paper demonstrates that certain genus two Mumford curves can be faithfully tropicalized in three-dimensional space, providing explicit conditions for when this is possible.

Contribution

It introduces a method to faithfully tropicalize genus two Mumford curves in dimension three under specific geometric conditions.

Findings

Faithful tropicalization possible for curves with disjoint cycles or sharing a short edge.

A map from the skeleton to the tropicalization of the Jacobian is constructed.

Conditions on cycle sharing determine tropicalization faithfulness.

Abstract

In the present paper, we investigate the question if the skeleton of a Mumford curve of genus two can be tropicalized faithfully in dimension three, i.e. if there exists an embedding of the curve in projective three space such that the tropicalization maps the skeleton of the curve isometrically to its image. Baker, Payne and Rabinoff showed that the skeleton of every analytic curve can be tropicalized faithfully. However the dimension of the ambient space in their proof can be quite large. We will define a map from the skeleton to the tropicalization of the Jacobian, which is an isometry on the cycles. It allows us to find principal divisors and simultanously to determine the retractions of their support on the skeleton which is necessary to calculate their tropicalization. It turns out that a Mumford curve of genus two whose cycles of its skeleton are either disjoint or share an edge of length at most half of the length of the cycles, can be tropicalized faithfully in dimension three.