Idealistic exponents: Tangent cone, ridge, characteristic polyhedra

TL;DR

This paper introduces the concept of characteristic polyhedra for idealistic exponents and provides an intrinsic geometric interpretation of tangent cones, directrix, and ridge over Spec(Z).

Contribution

It offers a novel geometric framework for understanding idealistic exponents through characteristic polyhedra, enhancing the interpretation of tangent cones, directrix, and ridge.

Findings

Defined characteristic polyhedra for idealistic exponents

Provided an intrinsic geometric interpretation of tangent cone, ridge, and directrix

Connected polyhedral data to properties of idealistic exponents

Abstract

We study Hironaka's idealistic exponents over $ \operatorname{Spec} ( \mathbb{Z} ) $. We give an idealistic interpretation of the tangent cone, the directrix, and the ridge. The main purpose is to introduce the notion of characteristic polyhedra of idealistic exponents and deduce from them intrinsic data on the idealistic exponent.