Whitney equisingularity for families of hypersurfaces in toric varieties

Abstract

In this paper, we establish conditions for a family $\{f_t\}$ of functions, with not necessarily isolated singularities, defined on a toric variety so that the associated family of hypersurfaces $\{f_t^{-1}(0)\}$ is Whitney equisingular. We work in the setting of toric varieties with arbitrary singular sets. This extends previous results by Eyral and Oka concerning families $\{F_t\}$ of functions in ${\mathbb{C}}^n$, with not necessarily isolated singularities, ensuring that the corresponding hypersurface family $\{F_t^{-1}(0)\}$ is Whitney equisingular.