Birational geometry of cluster algebras

TL;DR

This paper provides a geometric perspective on cluster varieties using blowups of toric varieties, offering new proofs, counterexamples, and insights into longstanding conjectures in cluster algebra theory.

Contribution

It introduces a geometric interpretation of cluster varieties, proves the Laurent phenomenon geometrically, and demonstrates the typical failure of the Fock-Goncharov dual basis conjecture.

Findings

Elementary geometric proof of the Laurent phenomenon

Counterexample to finite generation of certain upper cluster algebras

Most cases show the Fock-Goncharov dual basis conjecture is false

Abstract

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.