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 D be the scattering diagram of a cluster variety V (of either type– A or X ), and let S be a closed subset of (V^)^trop (R) —the ambient space of D . The set S is positive if the theta functions corresponding to the integral points of S and its N -dilations define an N -graded subalgebra of Γ (V, O_V) [x] . In particular, a positive set S defines a compactification of V through a Proj construction applied to the corresponding N -graded algebra. In this paper, we give a natural convexity notion for subsets of (V^)^trop (R), 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 (V^)^trop (R) or to check positivity of a given subset.