The mutation–qG-deformation correspondence for class TG del Pezzo surfaces

Let $P$ be a Fano polygon, and let $X_P$ be its associated toric del Pezzo surface. A del Pezzo surface with cyclic quotient singularities is class TG if it admits a qG-degeneration with reduced fibres to a normal toric del Pezzo surface. Fano polygons are mutation equivalent if they are connected by a sequence of combinatorial mutations, and del Pezzo surfaces with cyclic quotient singularities are qG-deformation equivalent if they are connected by a chain of qG-families over connected schemes. Let $\mathfrak{P}$ be the set of mutation equivalence classes of Fano polygons and let $\mathfrak{F}$ be the set of qG-deformation equivalence classes of locally qG-rigid class TG del Pezzo surfaces with cyclic quotient singularities.

Conjecture A. There is a one-to-one correspondence between $\mathfrak{P}$ and $\mathfrak{F}$. The correspondence sends $P$ to a generic qG-deformation of the toric surface $X_P$.

The authors prove that this assignment defines a surjective map $\mathfrak{P}\to\mathfrak{F}$, so the conjectural content is injectivity. This would describe the relationship between mutation classes of Fano polygons and the boundary structure of the moduli stack of del Pezzo surfaces.