On the Cohen–Macaulayness of some graded rings

Let (R,m) be a 1-dimensional Cohen-Macaulay local ring of multiplicity e and embedding dimension v ≥2 . Let B denote the blowing-up of R along m and let I be the conductor of R in B. Let x be a superficial element in m of degree 1 and I' = (I +xR)/xR . We assume that the length l(I') = 1 . This class of local rings contains the class of 1-dimensional Gorenstein local rings . In section 1, we prove that if the associated graded ring G = gr(R) is Cohen-Macaulay, then I is contained in m^s + xR , where s is the degree of the h-polynomial h(R) of R. In section 2, we give necessary and sufficient conditions for the Cohen-Macaulayness of G. These conditions are numerical conditions on the h-polynomial h(R) , particularly on its coefficients and the degree in comparison with the difference e − v . In section 3, we give some conditions for the Gorensteinness of G. In section 4, we give a characterisation (see 4.3) of numerical semigroup rings which satisfy the condition l(I') = 1