Cluster varieties and toric specializations of Fano varieties

Abstract

I state a conjecture asserting that for all generic klt Fano varieties X, there exists a generalised cluster variety U and a surjection from the set of torus charts on U to the set of toric specializations of X. I prove the conjecture in dimension 2 after work of Kasprzyk-Nill-Prince, Lutz, Hacking and Lai-Zhou. This confirms a deep and surprising structure to the classification of log del Pezzo surfaces first conjectured in work by Corti et al. In higher dimensions, I survey the evidence from the Fanosearch program, cluster structures for Grassmannians and flag varieties, and moduli spaces of conformal blocks. The paper is submitted for publication in a volume on the occasion of the 70th anniversary of V. V. Shokurov.