On the algebraic invariants of certain affine semigroup rings

Abstract

Let a and d be two linearly independent vectors in ${\mathbb {N}}^2$, over the field of rational numbers. For a positive integer $k \ge 2$, consider the sequence $a, a+d, \ldots , a+kd$ such that the affine semigroup $S_{a,d,k} = \langle a, a+d, \ldots , a+kd \rangle$ is minimally generated. We study the properties of affine semigroup ring $K[S_{a,d,k}]$ associated to this semigroup. We prove that $K[S_{a,d,k}]$ is always Cohen-Macaulay and it is Gorenstein if and only if $k=2$. For $k=2,3,4$, we explicitly compute the syzygies, the minimal graded free resolution and the Hilbert series of $K[S_{a,d,k}].$ We also give a minimal generating set for the defining ideal of $K[S_{a,d,k}]$ which is also a Gröbner basis. Consequently, we prove that $K[S_{a,d,k}]$ is Koszul. Finally, we prove that the Castelnuovo–Mumford regularity of $K[S_{a,d,k}]$ is 1 for any a, d, k.