A combinatorial extension of tropical cycles

TL;DR

This paper extends tropical intersection theory to complex spaces formed by glueing convex polyhedral cones, with applications to moduli spaces of tropical curves and their morphisms.

Contribution

It introduces a combinatorial framework for tropical cycles on glued polyhedral spaces, expanding the tools for studying tropical moduli spaces.

Findings

Defined tropical cycles on linear poic-complexes and poic-fibrations.

Developed pushforward maps for these tropical cycles.

Discussed clutching and forgetting morphisms in tropical moduli spaces.

Abstract

This article discusses a combinatorial extension of tropical intersection theory to spaces given by glueing quotients of partially open convex polyhedral cones by finitely many automorphisms. This extension is done in terms of linear poic-complexes and poic-fibrations, mainly motivated by the case of the moduli spaces of tropical curves of arbitrary genus and marking. We define tropical cycles of a linear poic-complex and of a poic-fibration, and discuss the pushforward maps in these situations. In the context of moduli spaces of tropical curves, we also discuss "clutching morphisms" and "forgetting the marking" morphisms. In a subsequent article we apply this framework to moduli spaces of discrete admissible covers and study the loci of tropical curves that appear as the source of a degree-$d$ discrete admissible cover of a genus-$h$ $m$-marked tropical curve, for fixed $d$, $h$ and $m$.