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In this article we explore compactifications of cluster varieties of finitetype in complex dimension two. Cluster varieties can be viewed as the spec of aring generated by theta functions and a compactification of such varieties canbe given by a grading on that ring, which can be described by positivepolytopes [17]. In the examples we exploit, the cluster variety can beinterpreted as the complement of certain divisors in del Pezzo surfaces. In thesymplectic viewpoint, they can be described via almost toric fibrations over$\R^2$ (after completion). Once identifying them as almost toric manifolds, onecan symplectically view them inside other del Pezzo surfaces. So we canidentify other symplectic compactifications of the same cluster variety, whichwe expect should also correspond to different algebraic compactifications. Bothviewpoints are presented here and several compactifications have theircorresponding polytopes compared. The finiteness of the cluster mutations areexplored to provide cycles in the graph describing monotone Lagrangian tori indel Pezzo surfaces connected via almost toric mutation [34].

Algebraic and symplectic viewpoint on compactifications of two-dimensional cluster varieties of finite type