CM defect and Hilbert functions of monomial curves

In this article we consider a semigroup ring R=K〚Γ〛 of a numerical semigroup Γ and study the Cohen–Macaulayness of the associated graded ring and the behaviour of the Hilbert function of R. We define a certain (finite) subset and prove that is Cohen–Macaulay if and only if . Therefore the subset is called the Cohen–Macaulay defect of . Further, we prove that if the degree sequence of elements of the standard basis of Γ is non-decreasing, then and hence is Cohen–Macaulay. We consider a class of numerical semigroups generated by 4 elements m0,m1,m2,m3 such that m1+m2=m0+m3—so called “balanced semigroups”. We study the structure of the Cohen–Macaulay defect of Γ and particularly we give an estimate on the cardinality for every r∈N. We use these estimates to prove that the Hilbert function of R is non-decreasing. Further, we prove that every balanced “unitary” semigroup Γ is “2-good” and is not “1-good”, in particular, in this case, is not Cohen–Macaulay. We consider a certain special subclass of balanced semigroups Γ. For this subclass we try to determine the Cohen–Macaulay defect using the explicit description of the standard basis of Γ; in particular, we prove that these balanced semigroups are 2-good and determine when exactly is Cohen–Macaulay.