Logarithmic Jacobian ideals, quasi-ordinary hypersurfaces and equisingularity

TL;DR

This paper explores the structure of quasi-ordinary hypersurface singularities by analyzing their Jacobian ideals and introduces logarithmic Jacobian ideals to better understand their equisingularity properties.

Contribution

It introduces the concept of logarithmic Jacobian ideals for quasi-ordinary hypersurfaces and compares them with monomial ideals to study equisingularity and normalization.

Findings

Logarithmic Jacobian ideals are monomial and relate to the normalization.

Comparison with monomial varieties provides insights into singularity structure.

Applications include criteria for equisingularity and normalization analysis.

Abstract

Let $(S, 0) \subset (\mathbb{C}^{d+1},0)$ be an irreducible germ of hypersurface. The germ $(S,0)$ is quasi-ordinary if $(S,0)$ has a finite projection to $(\mathbb{C}^d,0)$ which is unramified outside the coordinate hyperplanes. This implies that the normalization of $S$ is a toric singularity. One has also a monomial variety associated to $S$, which is a toric singularity with the same normalization, and with possibly higher embedding dimension. Since $(S,0)$ is quasi-ordinary, the extension of the Jacobian ideal of $S$ to the local ring of its normalization is a monomial ideal. We describe this monomial ideal by comparing it with the {\em logarithmic Jacobian ideals} of $S$ and of its associated monomial variety and we give some applications.