On the multiplicity of tangent cones of monomial curves

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.

Keywords

multiplicity, degree, associated graded ring, tangent cone, monomial curve, numerical semigroup, Betti numbers, initial ideal

2010 Mathematics Subject Classification

Primary 13A30. Secondary 13C40, 13D02, 13H10, 13H15, 13P10, 20M14.

Full Text (PDF format)

Received 19 November 2017

Received revised 22 June 2018

Accepted 4 July 2018

Published 3 May 2019