Universality for tropical and logarithmic maps

TL;DR

The paper demonstrates that spaces of tropical and logarithmic maps can realize all toric monoids and singularities, establishing a form of universality in their structure.

Contribution

It proves a universality theorem showing that all toric monoids appear in tropical and logarithmic map spaces, revealing their rich and varied singularity structures.

Findings

Every toric monoid appears in a space of tropical maps.

Spaces of logarithmic maps exhibit arbitrary toric singularities.

The cone over the 7-gon does not appear in rank 1 targets.

Abstract

We prove that every toric monoid appears in a space of maps from tropical curves to an orthant. It follows that spaces of logarithmic maps to Artin fans exhibit arbitrary toric singularities: a virtual universality theorem for logarithmic maps to pairs. The target rank depends on the chosen singularity: we show that the cone over the 7-gon never appears in a space of maps to a rank 1 target. We obtain similar results for tropical maps to affine space.