Mutations of Laurent Polynomials and Flat Families with Toric Fibers

TL;DR

This paper establishes a criterion for when two toric varieties can be fibers in a flat family over the projective line, linking Laurent polynomial mutations to deformations of toric varieties.

Contribution

It introduces a general criterion connecting Laurent polynomial mutations with deformations of associated toric varieties in flat families.

Findings

Certain birational transformations correspond to deformations between toric varieties.

A criterion for two toric varieties to be fibers in a flat family over the projective line.

Application to mutations of Laurent polynomials and their geometric interpretation.

Abstract

We give a general criterion for two toric varieties to appear as fibers in a flat family over the projective line. We apply this to show that certain birational transformations mapping a Laurent polynomial to another Laurent polynomial correspond to deformations between the associated toric varieties.