Cluster X-varieties at infinity

Mathematics – Algebraic Geometry

Scientific paper

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29 pages

Scientific paper

A positive space is a space with a positive atlas, i.e. a collection of rational coordinate systems with subtraction free transition functions. The set of positive real points of a positive space is well defined. We define a tropical compactification of the latter. We show that it generalizes the Thurston compactification of a Teichmuller space. The tropical boundary of a positive space is a sphere with a piecewise linear structure. Cluster X-varieties are positive spaces of rather special type. We define special completions of cluster X-varieties. They have a stratification whose strata are (affine closures of) cluster X-varieties. The original coordinate tori extend to coordinate affine spaces in the completion. We prove that, given a cyclic order of n+3 points on the projective line, the corresponding Deligne-Mumford moduli space is a blow up of the special completion of the cluster X-variety assigned to a root system A_n. The preimage of the affine closure of cluster X-variety itself is the complement to the Stasheff divisor in Deligne-Mumford moduli space. We prove Duality Conjectures from [FG2] for the cluster X-variety of type A_n. We define completions of Teichmuller spaces for surfaces with marked points at the boundary. The set of positive points of the special completion of the corresponding cluster X-variety is a part of the completion of the Teichmuller space.

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