Cluster varieties and toric specializations of Fano varieties

Abstract

I state a conjecture describing the set of toric specializations of a Fano variety with klt singularities. The conjecture asserts that for all generic Fano varieties X with klt singularities there exists a polarized cluster variety U and a surjection from the set of torus charts on U to the set of toric specializations of X. I outline the first steps of a theory of the cluster varieties that I use. In dimension 2, I sketch a proof of the conjecture after Kasprzyk–Nill–Prince, Lutz and Hacking by way of work of Lai–Zhou. This reveals a surprising structure to the classification of log del Pezzo surfaces that was first conjectured in [1]. In higher dimensions, I survey the evidence from the Fanosearch program, cluster structures for Grassmannians and flag varieties, and moduli spaces of conformal blocks.