On the algebraic invariants of certain affine semigroup algebras

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 sequencea, a+d, . . . , a+kdsuch that the affine semigroupSa,d,k=?a, a+d, . . . , a+kd?is minimally generated by this sequence. We study the properties of affinesemigroup algebra k[Sa,d,k] associated to this semigroup. We prove thatk [Sa,d,k] is always Cohen Macaulay and it is Gorenstein if and only if k= 2. Fork= 2,3,4, we explicitly compute the syzygies,minimal graded free resolution and Hilbert series of k[Sa,d,k]. We also give a minimal generating setand a Gr ?obner basis of the defining ideal of k[Sa,d,k]. Consequently, we prove that k[Sa,d,k] is Koszul. Finally, we prove that the Castelnuovo-Mumford regularity of k[Sa,d,k] is1 for any a, d, k.