On the Cohen-Macaulay property for quadratic tangent cones

TL;DR

This paper investigates when the tangent cone of a numerical semigroup ring with quadratic relations is Cohen-Macaulay, providing classifications for small embedding dimensions and linking Koszulness to quadratic Gr"obner bases.

Contribution

It proves Cohen-Macaulayness for tangent cones with up to five generators and characterizes non-Cohen-Macaulay cases explicitly, connecting algebraic properties.

Findings

For n<5, tangent cones are Cohen-Macaulay.

Explicit description of non-Cohen-Macaulay cases when n=5.

Equivalence of Koszulness and quadratic Gr"obner basis for n≤5.

Abstract

Let $H$ be an $n$-generated numerical semigroup such that its tangent cone $\operatorname{gr}_\mathfrak{m} K[H]$ is defined by quadratic relations. We show that if $n<5$ then $\operatorname{gr}_\mathfrak{m} K[H]$ is Cohen-Macaulay, and for $n=5$ we explicitly describe the semigroups $H$ such that $\operatorname{gr}_\mathfrak{m} K[H]$ is not Cohen-Macaulay. As an application we show that if the field $K$ is algebraically closed and of characteristic different from two, and $n\leq 5$ then $\operatorname{gr}_\mathfrak{m} K[H]$ is Koszul if and only if (possibly after a change of coordinates) its defining ideal has a quadratic Gr\"obner basis.