Birational geometry of cluster algebras. (English) Zbl 1322.14032
Summary: 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 D. Speyer’s example [“An infinitely generated upper cluster algebra”, Preprint, arXiv:1305.6867] of an upper cluster algebra which is not finitely generated, and show that the Fock-Goncharov dual basis conjecture is usually false.
MSC:
14E05 | Rational and birational maps |
13F60 | Cluster algebras |
14M25 | Toric varieties, Newton polyhedra, Okounkov bodies |
14J32 | Calabi-Yau manifolds (algebro-geometric aspects) |