Birational geometry of cluster algebras
By Mark Gross, P. Hacking, S. Keel. 2013
160 citations recorded by the source corpus.
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.
Subfields: Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/26395cf58d278e8effc98eeddee506efae09971b
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In Geometry and Topology
- Mark Gross
- P. Hacking
- S. Keel
- M. Kontsevich
- James Pascaleff
- Dmitry Tonkonog
- Bernd S Siebert
- M. Abouzaid
In Mathematical Physics
- Mark Gross
- M. Kontsevich
- James Pascaleff
- Dmitry Tonkonog
- P. Hacking
- S. Keel
- Bernd S Siebert
- M. Abouzaid
Citations in the corpus
- Canonical bases for cluster algebras (2014, background): “: Conjecture 0.10. The full Fock-Goncharov conjecture holds for Aprin if and only if the tropical space A∨ prin(R T) contains a full dimensional bounded polytope, convex in our sense. The examples of [GHK13], §7, show that for the full Fock-Goncharov conjecture to hold, we need to assume V has enoug”
- The wall-crossing formula and Lagrangian mutations (2017, background): “When some vectors among the {vi} coincide (which is the case for most del Pezzos in Table 1), the gluing construction of X∨ has to be modified as explained in [20].”