Nash multiplicities and isolated points of maximum multiplicity

Let $X$ be an algebraic variety defined over a field of characteristic zero, and let $\xi \in \underset{‾}{\mathrm{M}\mathrm{a}\mathrm{x}}\mathrm{m}\mathrm{u}\mathrm{l}\mathrm{t}(X)$ 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 $\underset{‾}{\mathrm{M}\mathrm{a}\mathrm{x}}\mathrm{m}\mathrm{u}\mathrm{l}\mathrm{t}(X)$.