On the defining equations of the tangent cone of a numerical semigroup ring

TL;DR

This paper investigates the algebraic structure of tangent cones of numerical semigroup rings, proving stability properties for large shifts and establishing bounds on their defining equations based on the semigroup's width.

Contribution

It demonstrates that for large shifts, the associated graded ring is Cohen–Macaulay with Betti numbers matching the original, and provides bounds on tangent cone equations depending on semigroup width.

Findings

Associated graded ring is Cohen–Macaulay for large shifts

Betti numbers of tangent cone match the original ring

Number of defining equations is bounded by semigroup width

Abstract

Let $\mathbf{a} = a_1 <\dots < a_r$ be a sequence of positive integers, and let $H_{\mathbf{a}}$ denote the semigroup generated by $a_1, \dots, a_r$. For an integer $k\geq 0$ we denote by $\mathbf{a}+k$ the shifted sequence $a_1 +k, \dots, a_r +k$. Fix a field $K$. We show that for all $k \gg 0$ the associated graded ring of the semigroup ring $K[H_{\mathbf{a}+k}]$ is Cohen--Macaulay and that it has the same Betti numbers as $K[H_{\mathbf{a}+k}]$ itself. As a consequence, we show that the number of defining equations of the tangent cone of a numerical semigroup ring is bounded by a value depending only on the width of the semigroup, where the width of a numerical semigroup is defined to be the difference of the largest and the smallest element in the minimal generating set of the semigroup. We also provide a conjectured upper bound of the above number of equations and we verify it in some cases.