Classification of Rank 2 Cluster Varieties

We classify rank $2$ cluster varieties (those for which the span of the rows of the exchange matrix is $2$-dimensional) according to the deformation type of a generic fiber $U$ of their ${\mathcal X}$-spaces, as defined by Fock and Goncharov [Ann. Sci. \'Ec. Norm. Sup\'er. (4) 42 (2009), 865-930]. Our approach is based on the work of Gross, Hacking, and Keel for cluster varieties and log Calabi-Yau surfaces. Call $U$ positive if $\dim[\Gamma(U,{\mathcal O}_U)] = \dim(U)$ (which equals 2 in these rank 2 cases). This is the condition for the Gross-Hacking-Keel construction [Publ. Math. Inst. Hautes \'Etudes Sci. 122 (2015), 65-168] to produce an additive basis of theta functions on $\Gamma(U,{\mathcal O}_U)$. We find that $U$ is positive and either finite-type or non-acyclic (in the usual cluster sense) if and only if the inverse monodromy of the tropicalization $U^{\rm trop}$ of $U$ is one of Kodaira's monodromies. In these cases we prove uniqueness results about the log Calabi-Yau surfaces whose tropicalization is $U^{\rm trop}$. We also describe the action of the cluster modular group on $U^{\rm trop}$ in the positive cases.