The leading ideal of a complete intersection of height two, Part II

Let (S,n) be a regular local ring and let I=(f,g) be an ideal in S generated by a regular sequence f,g of length two. Let R=S/I and m=n/I. As in [S. Goto, W. Heinzer, M.-K. Kim, The leading ideal of a complete intersection of height two, J. Algebra 298 (2006) 238–247], we examine the leading form ideal I∗ of I in the associated graded ring G=grn(S). If grm(R) is Cohen–Macaulay, we describe precisely the Hilbert series H(grm(R),λ) in terms of the degrees of homogeneous generators of I∗ and of their successive GCD's. If D=GCD(f∗,g∗) is a prime element of grn(S) that is regular on , we prove that I∗ is 3-generated and a perfect ideal. If htgrn(S)(f∗,g∗,h∗)=2, where h∈I is such that h∗ is of minimal degree in I∗∖(f∗,g∗)grn(S), we prove I∗ is 3-generated and a perfect ideal of grn(S), so grm(R)=grn(S)/I∗ is a Cohen–Macaulay ring. We give several examples to illustrate our theorems.