Compactifications of Cluster Varieties and Convexity

Gross–Hacking–Keel–Kontsevich [13] discuss compactifications of cluster varieties from positive subsets in the real tropicalization of the mirror. To be more precise, let ${\mathfrak{D}}$ be the scattering diagram of a cluster variety $V$ (of either type– ${\mathcal{A}}$ or ${\mathcal{X}}$), and let $S$ be a closed subset of $\left (V^\vee \right )^{\textrm{trop}} \left ({\mathbb{R}}\right )$—the ambient space of ${\mathfrak{D}}$. The set $S$ is positive if the theta functions corresponding to the integral points of $S$ and its ${\mathbb{N}}$-dilations define an ${\mathbb{N}}$-graded subalgebra of $\Gamma (V, \mathcal{O}_V){ [x]}$. In particular, a positive set $S$ defines a compactification of $V$ through a Proj construction applied to the corresponding ${\mathbb{N}}$-graded algebra. In this paper, we give a natural convexity notion for subsets of $\left (V^\vee \right )^{\textrm{trop}} \left ({\mathbb{R}}\right )$, called broken line convexity, and show that a set is positive if and only if it is broken line convex. The combinatorial criterion of broken line convexity provides a tractable way to construct positive subsets of $\left (V^\vee \right )^{\textrm{trop}} \left ({\mathbb{R}}\right )$ or to check positivity of a given subset.