A cluster variety $V$ admits, by definition, many charts of the form $(\mathbb{C}^*)^n \hookrightarrow V$. These charts do not always cover the variety of interest, but when they do, one could e.g. compute cohomologies using a Cech complex. For such purposes it would be useful to know:
Which algebraic varieties arise as the intersection of finitely many cluster charts?
Example: the space $V = \mathbb{C}^2 \setminus \{xy +1 = 0\} = \mathrm{Spec}\, \mathbb{C}[x,y,z, z^{-1}]/(z - xy - 1)$ has charts $\{x \ne 0, xy+1 \ne 0\}$ and $\{y \ne 0, xy+1 \ne 0\}$. Each is abstractly isomorphic to $(\mathbb{C}^*)^2$. $V$ is not the union of these charts, but in any case their intersection is abstractly isomorphic to $\mathbb{C}^* \times (\mathbb{C}^* \setminus 1)$.
