On the Hilbert function of the tangent cone of a monomial curve

TL;DR

This paper investigates the Hilbert function of the tangent cone of monomial curves, providing new conditions for non-decreasing behavior and improving existing results using invariants like the Apery set.

Contribution

It introduces a new sufficient condition for the Hilbert function of a numerical semigroup ring to be non-decreasing, independent of embedding dimension.

Findings

Established a new criterion for non-decreasing Hilbert functions

Improved previous results on Hilbert function behavior

Utilized semigroup invariants, especially the Apery set

Abstract

In this paper we study the Hilbert function of $\gr_{\mathfrak{m}}(R)$, when $R$ is a numerical semigroup ring or, equivalently, the coordinate ring of a monomial curve. In particular, we prove a sufficient condition for a numerical semigroup ring in order get a non-decreasing Hilbert function, without making any assumption on its embedding dimension; moreover, we show how this new condition allows to improve known results about this problem. To this aim we use certain invariants of the semigroup, with particular regard to its \Apery-set.