On the Cohen-Macaulay property for quadratic tangent cones

Abstract

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

arXiv comment21 pages; v2: minor edits, list of references updated; v3: added to Table 1 an example found by F. Strazzanti, and which is not in the published version