Laurent polynomials and deformations of non-isolated Gorenstein toric sigularities

Abstract

We establish a correspondence between one-parameter deformations of an affine Gorenstein toric pair $(X_P,\partial X_P)$, defined by a polytope $P$, and mutations of a Laurent polynomial $f$ with Newton polytope $\newt(f) = P$. For a Laurent polynomial $f$ in two variables, we construct a formal deformation of the three-dimensional Gorenstein toric pair $(X_{\newt(f)}, \partial X_{\newt(f)})$ over $\CC[[\bfTT_f]]$, where $\bfTT_f$ 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.