Physical Sciences and Mathematics Commons^™

Birational Geometry Of Cluster Algebras, Mark Gross, Paul Hacking, Sean Keel

Paul Hacking

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.