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Logarithmic Jacobian ideals, quasi-ordinary hypersurfaces and equisingularity

Pedro D. González Pérez

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.

Logarithmic Jacobian ideals, quasi-ordinary hypersurfaces and equisingularity

Abstract

Let be an irreducible germ of hypersurface. The germ is quasi-ordinary if has a finite projection to which is unramified outside the coordinate hyperplanes. This implies that the normalization of is a toric singularity. One has also a monomial variety associated to , which is a toric singularity with the same normalization, and with possibly higher embedding dimension. Since is quasi-ordinary, the extension of the Jacobian ideal of 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 and of its associated monomial variety and we give some applications.

Paper Structure

This paper contains 19 sections, 25 theorems, 115 equations.

Table of Contents

  1. On the Nash modification of an affine toric variety
  2. Equivariant embedding of an affine toric variety
  3. On the Nash modification of a toric singularity
  4. Quasi-ordinary hypersurface singularities
  5. Normalization of a quasi-ordinary singularity
  6. The semigroup of a quasi-ordinary hypersurface
  7. The associated toric variety $Z^\Gamma$
  8. The logarithmic Jacobian ideal of a quasi-ordinary hypersurface
  9. Logarithmic jacobian ideals
  10. The logarithmic Jacobian ideal of the quasi-ordinary hypersurface $(S,0)$.
  11. Description of the Jacobian ideal
  12. Overweight deformations
  13. The extension of the matrix of relations
  14. Description of the image of the Jacobian ideal of $S_p$
  15. An application
  16. ...and 4 more sections

Key Result

Lemma 1.1

(see T3, MR4792759) If the matrix $R^{m-d+1, \dots, p}$ is of rank $m-d$ then for any sequence $1\leq j_1 < \dots < j_d \leq m$ we have

Theorems & Definitions (59)

  • Lemma 1.1
  • proof
  • Proposition 1.3
  • proof
  • Definition 1.4
  • Remark 1.5
  • Proposition 1.6
  • proof
  • Example 1.7
  • Lemma 2.1
  • ...and 49 more