  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.
