Compactifications of Cluster Varieties and Convexity

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