Birational geometry of cluster algebras

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 [Spe13] of upper cluster algebras which are not finitely generated, and show that the Fock-Goncharov dual basis conjecture is usually false.Contents 1 Log Calabi-Yau varieties and a geometric motivation for cluster varieties 139 2 Review of the X and A cluster varieties 146 3 The geometry of cluster varieties 152 3.1 Elementary transformations. . . . . . . . . . . . . . . . . . . . . . . . . .152 3.2 The X -and A prin -cluster varieties up to codimension two. . . . . . . . .157 4 The A t and A prin cluster varieties as torsors 160 5 The X variety in the rank = 2 case 165 6 Examples of non-finitely-generated upper cluster algebras 170 7 Counterexamples to the Fock-Goncharov dual bases conjecture 171 References 173