On the Cohen-Macaulay property for quadratic tangent cones

Dumitru I. Stamate¹

¹University of Bucharest, Faculty of Mathematics and Computer Science, and Simion Stoilow Institute of Mathematics of the Romanian Academy, Research group of the project PN-II-RU-PD-2012-3-0656

The electronic journal of combinatorics, Tome 23 (2016) no. 3

Résumé

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 Gröbner basis.

Détail

DOI : 10.37236/5793

Classification : 13A30, 20M14
Mots-clés : numerical semigroup ring, tangent cone, Cohen-Macaulay, Koszul, \(G\)-quadratic, \(h\)-vector