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. 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, O U )] = dim(U ) (which equals 2 in these rank 2 cases). This is the condition for the [GHK15b]-construction to produce an additive basis of theta functions on Γ(U, 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 monodromy of the tropicalization U 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 trop . We also describe the action of the cluster modular group on U trop in the positive cases.

The canonical wall structure and intrinsic mirror symmetry

“…The present paper also provides the link to the algorithmic construction of walls [44], and to previous mirror constructions in two-dimensions [37] and for cluster varieties [38], also giving rise to rich combinatorial structures, see e.g. [22][23][24]62,63,74,75]. All these previous constructions generalize known and tested mirror constructions such as [12,13,35,51], and they have also been independently tested, see e.g.…”

Stable maps to Looijenga pairs

“…Looijenga pairs [79] were first systematically studied in relation with resolutions and deformations of elliptic surface singularities and with degenerations of K3 surfaces; see Friedman and Scattone [41]. More recently, Looijenga pairs have played an important role as two-dimensional examples for mirror symmetry; see Barrott [9], Bousseau [13], Gross, Hacking and Keel [53], Hacking and Keating [60], Mandel [81] and Yu [114; 115] and, for the theory of cluster varieties, Gross, Hacking and Keel [52], Mandel [82] and Zhou [117]. These new developments have had in return nontrivial applications to the classical geometry of Looijenga pairs; see Engel [38], Friedman [40] and Gross, Hacking and Keel [53; 54].…”

On a Nielsen–Thurston classification theory for cluster modular groups

“…The tropical compactification U is identified with the Thurston compactification of the Teichmüller space [15]. For an investigation of the action of the cluster modular group on U(Z t ), see [25].…”