Cluster algebras are Cox rings

TL;DR

This paper extends the connection between cluster algebras and Cox rings to include frozen vectors and partial compactifications, showing that theta bases provide bases of global sections under certain conditions.

Contribution

It generalizes previous results by incorporating frozen vectors and partial compactifications, and demonstrates the applicability of theta bases without skew-symmetrizability assumptions.

Findings

Extension of cluster algebra and Cox ring correspondence to frozen vectors

Theta bases form bases of global sections on compactified X-spaces

Results hold without skew-symmetrizability of exchange matrices

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.