CM defect and Hilbert function 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 G(Γ ) := grm(R) := ⊕n∈N mn/mn+1 and the behaviour of the Hilbert function HR of R.Wedefine a certain (finite) subset B(Γ ) ⊆ Γ and prove that G(Γ ) is Cohen–Macaulay if and only if B(Γ ) = ∅. Therefore the subset B(Γ ) is called the Cohen–Macaulay defect of G(Γ ). Further, we prove that if the degree sequence of elements of the standard basis of Γ is non-decreasing, then B(Γ ) = ∅ and hence G(Γ ) is Cohen–Macaulay. We consider a class of numerical semigroups Γ = Σ3 i=0 Nmi 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 B(Γ ) of Γ and particularly we give an estimate on the cardinality |B(Γ , r)| for every r ∈ N. We use these estimates to prove that the Hilbert function of R is nondecreasing. Further, we prove that every balanced ‘‘unitary’’ semigroup Γ is ‘‘2-good’’ and is not ‘‘1-good’’, in particular, in this case, G(Γ ) 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 B(Γ ) 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 G(Γ ) is Cohen–Macaulay.