On the defining equations of the tangent cone of a numerical semigroup ring

Abstract

Let a=a1<…<ar be a sequence of positive integers, and let Ha denote the semigroup generated by a1,…,ar. For an integer k≥0 we denote by a+k the shifted sequence a1+k,…,ar+k. Fix a field K. We show that for all k≫0 the tangent cone of the semigroup ring K[Ha+k] is Cohen–Macaulay and that it has the same Betti numbers as K[Ha+k] itself.As a consequence, we show that the number of defining equations of the tangent cone of a numerical semigroup ring is bounded by a value depending only on the width of the semigroup, where the width of a numerical semigroup is defined to be the difference of the largest and the smallest element in the minimal generating set of the semigroup. We also provide a conjectured upper bound of the above number of equations and we verify it in some cases.