On the last Hilbert-Samuel coefficient of isolated singularities (Q2350635)

The author proves that the last Hilbert-Samuel coefficient of an \(\mathbf m\)-primary ideal of a \(d\)-dimensional Cohen-Macaulay local ring \(R\) of characteristic zero is bounded by the geometric genus of \(X=\text{Spec}(R)\) (assumed to be an isolated singularity), provided that the associated graded ring of \(R\) with respect to \(I^n\) is Cohen-Macaulay for \(n\gg 0\). The ideals attaining the maximal value, are those for which the blow-up of the singularity centered at the ideal has only rational singularities. The one-dimensional case is studied separately in the last section.