I have expressed amazement before about the Laurent phenomenon for cluster algebras, a theorem of Fomin and Zelevinsky which I learned about from Lauren Williams. The paper “Birational Geometry of Cluster Algebras,” just posted by Mark Gross, Paul Hacking, and Sean Keel, seems extremely interesting on this point. They interpret the cluster transformations — which to an outsider look somewhat arbitrary — as elementary transforms (i.e. blow up a codim-2 thing and then blow down one of the exceptional loci thus created) on P^1-bundles on toric varieties. And apparently the Laurent phenomenon is plainly visible from this point of view. Very cool!

Experts are highly encouraged to weigh in below.

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