Tropical orbit spaces and the moduli spaces of elliptic tropical curves

TL;DR

This paper introduces tropical orbit spaces, explores their properties, and applies these concepts to simplify the proof of invariance of counts of elliptic tropical curves passing through given points.

Contribution

It defines tropical orbit spaces and their morphisms, and applies these to moduli spaces of elliptic tropical curves to streamline existing invariance proofs.

Findings

Weighted counts of elliptic tropical curves are independent of point configurations.

Tropical orbit spaces provide a framework for understanding invariance in tropical geometry.

Simplified proof of invariance of elliptic tropical curve counts.

Abstract

We give a definition of tropical orbit spaces and their morphisms. We show that, under certain conditions, the weighted number of preimages of a point in the target of such a morphism does not depend on the choice of this point. We equip the moduli spaces of elliptic tropical curves with a structure of tropical orbit space and, using our results on tropical orbit spaces, simplify the known proof of the fact that the weighted number of plane elliptic tropical curves of degree d with fixed j-invariant which pass through 3d-1 points in general position in $\RR^2$ is independent of the choice of a configuration of points.