Mutation–qG-deformation correspondence for Fano polygons and TG del Pezzo surfaces

A Fano polygon is a Fano polytope in the rank-two lattice setting, and two Fano polygons are mutation-equivalent when they are related by the combinatorial mutation operation. A del Pezzo surface with cyclic quotient singularities is of class TG if it admits a qG-deformation to a normal toric del Pezzo surface; locally qG-rigid means that its singularities are locally qG-rigid. Consider qG-deformation equivalence classes of such surfaces.

Conjecture A. There exists a bijective correspondence between the set of mutation-equivalence classes of Fano polygons and the set of qG-deformation equivalence classes of locally qG-rigid TG del Pezzo surfaces with cyclic quotient singularities.

This conjecture expresses the expected mirror-symmetric correspondence between mutation classes of Fano polygons and qG-deformation classes of associated del Pezzo surfaces. Its resolution status is not specified in the supplied source material.