On the multiplicity of tangent cones of monomial curves

Let $\mathrm{\Lambda }$ be a numerical semigroup, $\mathcal{C}\subseteq {\mathbb{A}}^n$ the monomial curve singularity associated to $\mathrm{\Lambda }$, and $\mathcal{T}$ its tangent cone. In this paper we provide a sharp upper bound for the least positive integer in $\mathrm{\Lambda }$ in terms of the codimension and the maximum degree of the equations of $\mathcal{T}$, when $\mathcal{T}$ is not a complete intersection. A special case of this result settles a question of J. Herzog and D. Stamate.