Cluster varieties and toric specializations of Fano varieties

TL;DR

The paper conjectures a deep link between Fano varieties and cluster varieties, proving it in dimension 2 and providing evidence and surveys for higher dimensions, revealing new structural insights into Fano classification.

Contribution

It introduces a conjecture connecting Fano varieties with cluster varieties and proves it for dimension 2, advancing understanding of Fano classification structures.

Findings

Confirmed the conjecture for dimension 2 Fano varieties.

Identified a surjection from cluster variety torus charts to toric specializations.

Surveyed evidence supporting the conjecture in higher dimensions.

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.