The numerical monoid description of flag and ordinary higher Nash blowups

Let $\b X$ be a formal curve with associated numerical monoid

$$S=\{0=s_0<s_1<\cdots\}.$$

For a numerical monoid, let the associated numerical monoid of a blowup be the numerical monoid corresponding to its complete local algebra. Numerical monoid description. The associated numerical monoid of $\operatorname{Nash}_n(\b X)$ is generated by

$$s_m-s_l,\qquad m>n,\quad l\leq n,$$

and that of $\operatorname{fNash}_n(\b X)$ is generated by

$$s_m-s_l,\qquad m>l,\quad l\leq n.$$

This gives an explicit description of the ordinary and flag higher Nash blowups for formal curves in terms of the associated numerical monoid. The supplied context illustrates the distinction between these constructions, but does not state whether the result has an independent resolution status.