2019/12/30 by Cheung, Man-Wai, Magee, Timothy, Chávez, Alfredo Nájera · 1 citation
#Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.1912.13052
In [GHKK18], Gross-Hacking-Keel-Kontsevich discuss compactifications of cluster varieties from "positive subsets" in the real tropicalization of the mirror. To be more precise, let \mathfrakD be the scattering diagram of a cluster variety V (of either type -- A or X), and let S be a closed subset of (V^\vee)trop(ℝ) -- the ambient space of \mathfrakD. 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, OV)[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 \mathfrakD, 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 \mathfrakD, or to check positivity of a given subset.