Whitney equisingularity for families of hypersurfaces in toric varieties

TL;DR

This work extends Whitney equisingularity results from $\mathbb{C}^n$ to hypersurface families defined on general toric varieties, including non-isolated singularities. By introducing essential non-compact faces and a toric-adapted notion of local tameness, the authors define admissible families that keep the Newton boundary invariant and ensure non-degeneracy across strata. The main result proves that admissible toric families yield Whitney equisingular hypersurface families, achieved via a canonical stratification and a rigorous verification of Whitney $(b)$-regularity through Newton-polyhedron techniques and Curve Selection arguments. The findings bridge toric geometry, non-degeneracy theory, and equisingularity, with implications for studying parameterized singularities in toric settings and related determinantal structures.

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.

Figures (4)

• Figure 1: Polyhedral cone in $\mathbb{R}^3$
• Figure 2: Newton polyhedron on $\mathbb C^2$
• Figure 3: Newton polyhedron on $X(S_\sigma)$
• Figure 4: Essential non-compact face

Theorems & Definitions (30)

• Theorem A
• Theorem B: see Theorem \ref{['mainteo']}
• Definition 1
• Definition 2
• Definition 3
• Theorem 4
• Remark 5
• Definition 6
• Example 1
• Remark 7