Nash multiplicity sequences and Hironaka's order function

TL;DR

This paper links Hironaka's order function in resolution of singularities to Nash multiplicity sequences, providing a geometric interpretation and potential refinements for positive characteristic cases.

Contribution

It demonstrates that Hironaka's order function can be understood through Nash multiplicity sequences, offering new insights into its geometric meaning.

Findings

Hironaka's order function relates to Nash multiplicity sequences

The function has an intrinsic geometric interpretation

Potential for refined resolution techniques in positive characteristic

Abstract

When $X$ is a $d$-dimensional variety defined over a field $k$ of characteristic zero, a constructive resolution of singularities can be achieved by successively lowering the maximum multiplicity via blow ups at smooth equimultiple centers. This is done by stratifying the maximum multiplicity locus of $X$ by means of the so called {\em resolution functions}. The most important of these functions is what we know as {\em Hironaka's order function in dimension $d$}. Actually, this function can be defined for varieties when the base field is perfect; however if the characteristic of $k$ is positive, the function is, in general, too coarse and does not provide enough information so as to define a resolution. It is very natural to ask what the meaning of this function is in this case, and to try to find refinements that could lead, ultimately, to a resolution. In this paper we show that Hironaka's order function in dimension $d$ can be read in terms of the {\em Nash multiplicity sequences} introduced by Lejeune-Jalabert. Therefore, the function is intrinsic to the variety and has a geometrical meaning in terms of its space of arcs.