Quasi-ordinary hypersurfaces, multiplier ideals and local tropicalizations

TL;DR

This paper characterizes multiplier ideals and jumping numbers of quasi-ordinary hypersurfaces using toroidal resolutions, Newton polyhedra, and local tropicalizations, linking algebraic and combinatorial structures.

Contribution

It introduces a method to describe multiplier ideals of quasi-ordinary hypersurfaces via Newton polyhedra and tropical geometry, extending known monomial ideal results.

Findings

Multiplier ideals are expressed through Newton polyhedra.

Multiplier ideals are generalized monomial ideals with semi-root sequences.

Local tropicalization forms a fan determined by the hypersurface's topological type.

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 $(D, 0) \subset (\mathbb{C}^{d+1}, 0)$.