Given a Laurent polynomial with reflexive Newton polytope, we give a criterion ensuring that it gives rise to a deformation family with smooth general fibre. One-parameter deformations in this family correspond to mutations of the Laurent polynomial.

In dimension three, the smooth fibres arising from this construction include all 98 deformation families of very ample smooth Fano threefolds. This is an upcoming result that generalises the published work Laurent Polynomials and Deformations of Non-Isolated Gorenstein Toric Singularities.

Building on that framework, I will also outline evidence towards a conjecture of Corti–Filip–Petracci predicting a one-to-one correspondence between 0-mutable Laurent polynomials and the smoothing deformation components of three-dimensional affine Gorenstein toric varieties.