Classification of Rank 2 Cluster Varieties

Abstract

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 X-spaces, as defined by Fock and Goncharov [Ann. Sci. Éc. Norm. Supér. (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[Γ(U, OU)]=dim(U) (which equals 2 in these rank 2 cases). This is the condition for the Gross-Hacking-Keel construction [Publ. Math. Inst. Hautes Études Sci. 122 (2015), 65-168] to produce an additive basis of theta functions on Γ(U, OU). 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 Uᵗʳᵒᵖ of U is one of Kodaira's monodromies. In these cases, we prove uniqueness results about the log Calabi-Yau surfaces whose tropicalization is Uᵗʳᵒᵖ. We also describe the action of the cluster modular group on Uᵗʳᵒᵖ in the positive cases.