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A combinatorial extension of tropical cycles

Diego A. Robayo Bargans

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 $m$.

A combinatorial extension of tropical cycles

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- discrete admissible cover of a genus- -marked tropical curve, for fixed , and .

Paper Structure

This paper contains 27 sections, 48 theorems, 135 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Partially open integral cones.
  4. Discrete graphs.
  5. Some categories of graphs and their structures.
  6. Tropical curves and their moduli spaces.
  7. Poic-complexes and their cycles
  8. Poic-complexes.
  9. Linear poic-complexes and Minkowski weights.
  10. Moduli space of rational tropical curves.
  11. Subdivisions and tropical cycles.
  12. Pushforwards.
  13. Relation to tropical intersection theory.
  14. Poic-fibrations and their cycles
  15. Poic-spaces.
  16. ...and 12 more sections

Key Result

Lemma 2.2.5

Let $G$ be a discrete graph, with $e_1,e_2\in E(G))$ two different incident edges. Then $\left(G/e_1\right)/e_2$ is naturally bijective to $\left(G/e_2\right)/e_1$.

Figures (11)

  • Figure 1: The theta graph
  • Figure 2: Skeleton of $\mathbb{G}_{2,1}$ without automorphisms of objects.
  • Figure 3: Examples of cones of metrics.
  • Figure 4: Example of a poic-fibration
  • Figure 5: Depiction of poic-spaces of Example \ref{['ex: working example']}.
  • ...and 6 more figures

Theorems & Definitions (164)

  • Definition 2.1.1
  • Definition 2.1.2
  • Remark
  • Definition 2.1.3
  • Remark
  • Definition 2.1.5
  • Example 2.3
  • Example 2.4
  • Definition 2.2.1
  • Example 2.5
  • ...and 154 more