Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Quasi-ordinary hypersurfaces, multiplier ideals and local tropicalizations

Pedro D. González Pérez, Miguel Robredo Buces

TL;DR

This work develops a Howald-type framework for multiplier ideals and jumping numbers of irreducible quasi-ordinary hypersurface singularities by leveraging a toroidal embedded resolution built from a complete sequence of semi-roots. Multiplier ideals are described as generalized monomial ideals in semi-root coordinates, with jumping numbers extracted from finite Newton polyhedra attached to the total transform and the boundary divisors of the resolution; log-discrepancies play a central role in these criteria. A key innovation is the fusion of toroidal resolution data with local tropicalization: Trop$(Y)$ is shown to be the support of a fan determined by the embedded topological type, enabling a tropical reinterpretation of multiplier-ideal generators and jumping numbers. The results yield a finite, computable scheme for explicitly determining multiplier ideals and their jumps in higher dimensions, and they illuminate the interplay between resolution geometry, quasi-monomial valuations, and tropical geometry.

Abstract

In this paper we describe the multiplier ideals and jumping numbers associated with an irreducible germ of quasi-ordinary hypersurface $(D, 0) \subset (\mathbb{C}^{d+1}, 0)$ by using a toroidal embedded resolution. The approach is motivated by Howald's description of the multiplier ideals of monomial ideals. We show that the multiplier ideals of $D$ can be expressed in terms of a finite sequence of Newton polyhedra associated with the total transform of $D$ in the toroidal resolution process. We prove that the multiplier ideals are generalized monomial ideals with respect to a complete sequence of semi-roots. This is a finite sequence of functions which determines a system of generators of the semigroup of the quasi-ordinary hypersurface. We express these results in terms of the local tropicalization associated with the embedding of $\mathbb{C}^{d+1}$ defined by this sequence. We prove that the local tropicalization is the support of a fan of the lattice $\mathbb{Z}^{d+g+1}$, which is determined by the embedded topological type of $(D, 0) \subset (\mathbb{C}^{d+1}, 0)$.

Quasi-ordinary hypersurfaces, multiplier ideals and local tropicalizations

TL;DR

This work develops a Howald-type framework for multiplier ideals and jumping numbers of irreducible quasi-ordinary hypersurface singularities by leveraging a toroidal embedded resolution built from a complete sequence of semi-roots. Multiplier ideals are described as generalized monomial ideals in semi-root coordinates, with jumping numbers extracted from finite Newton polyhedra attached to the total transform and the boundary divisors of the resolution; log-discrepancies play a central role in these criteria. A key innovation is the fusion of toroidal resolution data with local tropicalization: Trop is shown to be the support of a fan determined by the embedded topological type, enabling a tropical reinterpretation of multiplier-ideal generators and jumping numbers. The results yield a finite, computable scheme for explicitly determining multiplier ideals and their jumps in higher dimensions, and they illuminate the interplay between resolution geometry, quasi-monomial valuations, and tropical geometry.

Abstract

In this paper we describe the multiplier ideals and jumping numbers associated with an irreducible germ of quasi-ordinary hypersurface by using a toroidal embedded resolution. The approach is motivated by Howald's description of the multiplier ideals of monomial ideals. We show that the multiplier ideals of can be expressed in terms of a finite sequence of Newton polyhedra associated with the total transform of in the toroidal resolution process. We prove that the multiplier ideals are generalized monomial ideals with respect to a complete sequence of semi-roots. This is a finite sequence of functions which determines a system of generators of the semigroup of the quasi-ordinary hypersurface. We express these results in terms of the local tropicalization associated with the embedding of defined by this sequence. We prove that the local tropicalization is the support of a fan of the lattice , which is determined by the embedded topological type of .
Paper Structure (14 sections, 34 theorems, 148 equations, 1 figure)

This paper contains 14 sections, 34 theorems, 148 equations, 1 figure.

Table of Contents

  1. Basic notions about multiplier ideals and jumping numbers
  2. Basic notions on toric geometry and Newton polyhedra
  3. Toroidal embeddings
  4. Quasi-ordinary hypersurface singularities
  5. Expansions in term of the semi-roots
  6. Quasi-ordinary hypersurfaces relative to a toric base
  7. Some combinatorial facts
  8. Toroidal embedded resolutions of a quasi-ordinary hypersurface
  9. Quasi-monomial valuations of the toroidal embedded resolution
  10. Generating sequences of the quasi-monomial valuations
  11. Log-discrepancies of exceptional divisors
  12. Multiplier ideals of a quasi-ordinary hypersurface
  13. The local tropicalization defined by the sequence semi-roots
  14. The multiplier ideals and the local tropicalization

Key Result

Theorem 1

Take $\xi \in \mathbb Q_{\geq 0}$ and $h \in \mathcal{O}_{\mathbb C^{d+1},0}$. The following conditions are equivalent:

Figures (1)

  • Figure 1: A sketch representing the local tropicalization of $Y$.

Theorems & Definitions (90)

  • Theorem 1
  • Theorem 2
  • Definition 1.1
  • Lemma 1.2
  • Lemma 1.3
  • Remark 1.4
  • Proposition 2.1
  • proof
  • Remark 2.3
  • Definition 2.4
  • ...and 80 more