Laurent polynomials and deformations of non-isolated Gorenstein toric sigularities

We establish a correspondence between one-parameter deformations of an affine Gorenstein toric pair $(X_P,\mathrm{\partial }{X}_P)$, defined by a polytope $P$, and mutations of a Laurent polynomial $f$ with Newton polytope $\text{newt}(f)=P$. For a Laurent polynomial $f$ in two variables, we construct a formal deformation of the three-dimensional Gorenstein toric pair $(X_{\text{newt}(f)},\mathrm{\partial }{X}_{\text{newt}(f)})$ over $\text{CC}[[{\text{bfTT}}_f]]$, where ${\text{bfTT}}_f$ is the set of deformation parameters arising from mutations. The general fibre of this deformation is smooth if and only if $f$ is $0$-mutable. The Kodaira–Spencer map of the constructed deformation is injective, and if $f$ is maximally mutable, then the deformation cannot be nontrivially extended to a larger smooth base space.