Tropicalization of curve arrangement complements and arroids

TL;DR

This paper introduces arroids as a new combinatorial structure to encode intersection properties of curve arrangements and explores their tropicalization, linking combinatorics, topology, and tropical geometry.

Contribution

It defines arroids, establishes their role in tropicalizing curve arrangement complements, and provides criteria for cohomology computation and maximality in tropical geometry.

Findings

Arroids encode intersection properties of curve arrangements.

Tropicalization of arrangement complements is determined by arroids.

Conditions are given for cohomology computability and maximality in tropical geometry.

Abstract

We define arroids as an abstract axiom set encoding the intersection properties of arrangements of curves. The tropicalization of the complement of arrangement of curves meeting pairwise transversely is shown to be determined by the associated arroid. We give conditions for when the cohomology of the complement of an arrangement is computable using tropical cohomology, and we give criteria for when the complement is a maximal variety in terms of tropical geometry.