Compactifications of Cluster Varieties and Convexity. (6th April 2021) - Cheung, Man-Wai; Magee, Timothy; Nájera Chávez, Alfredo
Abstract: 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 $\left (V^\vee \right )^{trop} \left ({\mathbb {R}}\right )$ —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 $Γ (V, \mathcal {O}_V){ [x]}$ . 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 $\left (V^\vee \right )^{trop} \left ({\mathbb {R}}\right )$, 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 $\left (V^\vee \right )^{trop} \left ({\mathbb {R}}\right )$ or to check positivity of a given subset.