The asymptotic Samuel function and invariants of singularities

TL;DR

This paper introduces the Samuel slope, an invariant derived from the asymptotic Samuel function, to analyze singularities in algebraic varieties and connect it with existing invariants used in resolution algorithms.

Contribution

It defines the Samuel slope for Noetherian local rings and explores its properties, linking it to invariants in singularity resolution.

Findings

Samuel slope relates to resolution invariants

Properties of Samuel slope in singular rings

Connection with algorithmic resolution methods

Abstract

The asymptotic Samuel function generalizes to arbitrary rings the usual order function of a regular local ring. In this paper, we use this function to introduce the notion of the Samuel slope of a Noetherian local ring, and we study some of its properties. In particular, we focus on the case of a local ring at singular point of a variety, and, among other results, we prove that the Samuel slope of these rings is related to some invariants used in algorithmic resolution of singularities.