Blow-up rings and rational singularities
Let \(A\) be a normal local ring which is essentially of finite type over a field of characteristic zero. Let \(I\) be an ideal of \(A\) such that the Rees algebra \(R_A(I)\) is Cohen-Macaulay and normal. This paper investigates when \(R_A(I)\) has rational singularities. It is shown that if \(A\) has rational singularities and \(R_A(I)\) satisfies Serre's condition \((S_{d-1})\) where \(d=\dim A\), then \(R_A(I)\) has rational singularities. It is also shown that if \(A\) is a regular local ring of dimension \(d\) with \(\text{Proj} R_a(I)\) normal and with the adjoint of \(I_{d-2}\) equal to \(A\), then \(\text{Proj} R_A(I)\) has rational singularities. However, it is noted that is \(A=k[x,y,z]_{(x,y,z)}\) where \(k\) is a field of characteristic zero and \(I=pA\) where \(p\) is the prime ideal of the monomial curve \(x=t^{11}\), \(y=t^{14}\), and \(z=t^{15}\), then \(R_A(I)\) is normal Cohen-Macaulay, but \(R_A(I)\) does not have rational singularities. This example also provides a counterexample to a conjecture of Lipman: Let \(A\) be a regular local ring and \(I\) and \(J\) proper integrally closed ideals of \(A\). If \(R_A(IJ)\) is normal and Cohen-Macaulay, then \(IR_A(I^eJ^e)\) is Cohen-Macaulay for \(e\gg 0\).
scientific article; zbMATH DE number 1295394
scientific article