Idealistic exponents: Tangent cone, ridge, characteristic polyhedra

Abstract

We study Hironaka's idealistic exponents over $\mathrm{Spec}(\mathbb{Z})$. 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.

arXiv comment41 pages; reworked some parts of the article: corrected and clarified some steps; added new results on the codimension of the ridge as an upper semi-continuous invariant; introduced a new idealistic exponent refining the idealistic version of the tangent cone; updated the references; changed the title