We give a geometric interpretation of cluster varieties in terms of blowups
of toric varieties. This enables us to provide, among other results, an
elementary geometric proof of the Laurent phenomenon for cluster algebras (of
geometric type), extend Speyer's example of an upper cluster algebra which is
not finitely generated, and show that the Fock-Goncharov dual basis conjecture
is usually false.Comment: 50 pages, to appear in Algebraic Geometr
