Nash multiplicities and isolated points of maximum multiplicity

TL;DR

This paper introduces a criterion based on arc invariants and Nash multiplicity sequences to determine when a point of maximum multiplicity on an algebraic variety is isolated, enhancing understanding of singularities.

Contribution

It provides a new arc-based criterion using Nash multiplicity invariants to identify isolated points of maximum multiplicity on algebraic varieties.

Findings

Criterion for isolation of maximum multiplicity points using arcs

Characterization via Nash multiplicity sequence invariants

Applicable to algebraic varieties over characteristic zero fields

Abstract

Let $X$ be an algebraic variety defined over a field of characteristic zero, and let $\xi \in \mathrm{\underline{Max}\; mult}(X)$ be a point in the closed subset of maximum multiplicity of $X$. We provide a criterion, given in terms of arcs, to determine whether $\xi $ is isolated in $\mathrm{\underline{Max}\; mult}(X)$. More precisely, we use invariants of arcs derived from the Nash multiplicity sequence to characterize when $\xi $ is an isolated point in $\mathrm{\underline{Max}\; mult}(X)$.