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Birational geometry of cluster algebras

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arxiv 1309.2573 v2 pith:2TGM6WXD submitted 2013-09-10 math.AG

Birational geometry of cluster algebras

classification math.AG
keywords clustergeometricalgebrasvarietiesalgebrabasisbirationalblowups
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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.

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