Nash multiplicities and resolution invariants

TL;DR

This paper explores how the Nash multiplicity sequence, originally defined for hypersurface germs, can be used to compute invariants relevant to the algorithmic resolution of singularities, building on previous generalizations.

Contribution

It demonstrates the application of the Nash multiplicity sequence to compute invariants in the context of resolving singularities, extending prior theoretical work.

Findings

The Nash multiplicity sequence can be effectively used to compute resolution invariants.

Generalization of the sequence allows for broader application in singularity resolution.

Provides a method to connect multiplicity sequences with algorithmic resolution processes.

Abstract

The Nash multiplicity sequence was defined by M. Lejeune-Jalabert as a non-increasing sequence of integers attached to a germ of a curve inside a germ of a hypersurface. M. Hickel generalized this notion and described a sequence of blow ups which allows us to compute it and study its behavior. In this paper, we show how this sequence can be used to compute some invariants that appear in algorithmic resolution of singularities.