On the multiplicity of tangent cones of monomial curves

TL;DR

This paper establishes a sharp upper bound for the smallest positive element in a numerical semigroup associated with a monomial curve, relating it to the tangent cone’s properties, and addresses a question posed by Herzog and Stamate.

Contribution

It provides a new upper bound for the minimal element in the semigroup based on tangent cone invariants, advancing understanding of monomial curve singularities.

Findings

Derived a sharp upper bound for the least positive integer in the semigroup.

Connected the bound to the codimension and maximum degree of tangent cone equations.

Resolved a specific question posed by Herzog and Stamate.

Abstract

Let $\Lambda$ be a numerical semigroup, $\mathcal{C}\subseteq \mathbb{A}^n$ the monomial curve singularity associated to $\Lambda$, and $\mathcal{T}$ its tangent cone. In this paper we provide a sharp upper bound for the least positive integer in $\Lambda$ 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.