Tropical fans and the moduli spaces of tropical curves

TL;DR

This paper rigorously defines tropical fans and their morphisms, establishing a foundational intersection theory and applying it to moduli spaces of rational tropical curves, proving invariance of certain enumerative counts.

Contribution

It introduces a rigorous definition of tropical fans and morphisms, and develops a tropical intersection theory with applications to moduli spaces of rational tropical curves.

Findings

Number of inverse images under tropical fan morphisms is independent of the point chosen.

Provides new proofs for the invariance of counts of rational tropical curves through given points.

Establishes a foundational framework for tropical intersection theory.

Abstract

We give a rigorous definition of tropical fans (the "local building blocks for tropical varieties") and their morphisms. For such a morphism of tropical fans of the same dimension we show that the number of inverse images (counted with suitable tropical multiplicities) of a point in the target does not depend on the chosen point - a statement that can be viewed as the beginning of a tropical intersection theory. As an application we consider the moduli spaces of rational tropical curves (both abstract and in some R^r) together with the evaluation and forgetful morphisms. Using our results this gives new, easy, and unified proofs of various tropical independence statements, e.g. of the fact that the numbers of rational tropical curves (in any R^r) through given points are independent of the points.