On the integral closures of certain ideals generated by regular sequences (Q1582750)

The goal of the paper under review is to introduce a method for computing the integral closures of certain ideals generated by regular sequences which are generically good. More precisely, let \(A=k[[X,Y,Z]]\) be the formal power series ring over a field \(k\), and \(\alpha\), \(\alpha'\), \(\beta\), \(\beta'\), \(\gamma\), \(\gamma'\) be arbitrary positive integers. Set \(a=Z^{\gamma+\gamma'}-X^{\alpha'}Y^{\beta'}\), \(b=X^{\alpha+\alpha'}-Y^{\beta}Z^{\gamma'}\), and \(c=Y^{\beta+\beta'}-X^{\alpha}Z^{\gamma}\). Let \(J=(a,b)A\) and \(I\) be the ideal of \(A\) generated by \(a\), \(b\) and the set of the \(X^iZ^jc\) with \(i,j\geq 0\) and \(i/\alpha'+j/\gamma'\geq 1\). Then, the author shows that the integral closure \(\overline{J^n}\) is equal to \(J^{n-1}I\) for all \(n\geq 1\) and the Rees algebra \(R(\overline{J})\) is a Cohen-Macaulay ring. The key results for the proof are: The associated primes in \(A\) of \(A/J\) are \((a,b,c)A\) and \((X,Y)A\), and \(A/J^nI\) is Cohen-Macaulay for all \(n\geq 0\).