On the algebraic invariants of certain affine semigroup rings

Abstract

Let a and d be two linearly independent vectors in N2, over the field of rational numbers. For a positive integer k≥2, consider the sequence a,a+d,…,a+kd such that the affine semigroup Sa,d,k=⟨a,a+d,…,a+kd⟩ is minimally generated. We study the properties of affine semigroup ring K[Sa,d,k] associated to this semigroup. We prove that K[Sa,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[Sa,d,k]. We also give a minimal generating set for the defining ideal of K[Sa,d,k] which is also a Gröbner basis. Consequently, we prove that K[Sa,d,k] is Koszul. Finally, we prove that the Castelnuovo–Mumford regularity of K[Sa,d,k] is 1 for any a, d, k.