Cluster algebras are Cox rings

Abstract

It was recently shown by Gross, Hacking, and Keel that, in the absence of frozen indices, a cluster A-variety with generic coefficients is the universal torsor of the corresponding cluster X-variety with corresponding coefficients. We extend this to allow for frozen vectors and corresponding partial compactifications of the A- and X-spaces. When certain assumptions are satisfied, we conclude that the theta bases of Gross-Hacking-Keel-Kontsevich give bases of global sections for every line bundle on the leaves of the partially compactified X-space. We note that our arguments work without assuming that the exchange matrix is skew-symmetrizable.