Abstract
Gross, Hacking, and Keel have shown that, in the absence of frozen indices, a cluster \(\mathcal {A}\)-variety with generic coefficients is the universal torsor of the corresponding cluster \(\mathcal {X}\)-variety with corresponding coefficients. We extend this to allow for (partial) compactifications of the \(\mathcal {A}\)- and \(\mathcal {X}\)-spaces, using frozen vectors to capture the data of the boundary divisors. This works even without assuming that the exchange matrix is skew-symmetrizable. When certain assumptions are satisfied, we conclude that the theta bases on the \(\mathcal {A}\)-space yield additive bases of global sections for every line bundle on the leaves of the partially compactified \(\mathcal {X}\)-space.
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Acknowledgements
The author would like to thank Man-Wai Cheung for helpful discussions and for suggesting the project that motivated this work, as well as Sean Keel for helpful feedback. I would also like to thank the anonymous referee for helpful expository suggestions.
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The author was supported by the National Science Foundation RTG Grant DMS-1246989.
