The asymptotic Samuel functions and applications to resolution of singularities techniques

TL;DR

This paper introduces the Samuel slope and refined Samuel slope, invariants derived from the asymptotic Samuel function, which characterize regularity in local rings and relate to resolution of singularities techniques.

Contribution

It defines the Samuel slope and refined Samuel slope, establishing their role in characterizing regularity and linking them to resolution of singularities.

Findings

Samuel slope characterizes regularity in local excellent rings

Refined Samuel slope provides a more detailed invariant

Connections established between these invariants and resolution techniques

Abstract

We use the asymptotic Samuel function to define the Samuel slope of a Noetherian local ring, and we prove that it characterizes regularity in the case of local excellent rings. In addition, we introduce a second invariant that refines the Samuel slope using generic linear cuts, the refined Samuel slope. We show that both, the Samuel slope and the refined Samuel slope, are connected to invariants coming from resolution of singularities.