On the algebraic invariants of certain affine semigroup rings

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 S<inf>a</inf><inf>,</inf><inf>d</inf><inf>,</inf><inf>k</inf>= ⟨ a, a+ d, … , a+ kd⟩ is minimally generated. We study the properties of affine semigroup ring K[S<inf>a</inf><inf>,</inf><inf>d</inf><inf>,</inf><inf>k</inf>] associated to this semigroup. We prove that K[S<inf>a</inf><inf>,</inf><inf>d</inf><inf>,</inf><inf>k</inf>] 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<inf>a</inf><inf>,</inf><inf>d</inf><inf>,</inf><inf>k</inf>]. We also give a minimal generating set for the defining ideal of K[S<inf>a</inf><inf>,</inf><inf>d</inf><inf>,</inf><inf>k</inf>] which is also a Gröbner basis. Consequently, we prove that K[S<inf>a</inf><inf>,</inf><inf>d</inf><inf>,</inf><inf>k</inf>] is Koszul. Finally, we prove that the Castelnuovo–Mumford regularity of K[S<inf>a</inf><inf>,</inf><inf>d</inf><inf>,</inf><inf>k</inf>] is 1 for any a, d, k.