Ideals having a one-dimensional fiber cone (Q2741193)

Let \(I\) be a regular ideal in a Noetherian local ring \((R, \mathcal M)\) having a principal reduction in \(R\). The authors want to see how some properties of \(I^n\) are reflected to the associated graded algebra \(G(I)\) and to the fiber cone \(F(I)\). One of the aims of the paper is to construct several examples which show some peculiarities of the involved notions. NEWLINENEWLINENEWLINEThe first point of investigation is the relation between the postulation number of \(F(I)\) and the reduction number \(r(I)\) of \(I\). The authors introduce the postulation number \(\text{fp}(I)\) of \(I\) (i.e. the largest integer \(n\) such that the Hilbert function and the Hilbert polynomials of \(F(I)\) are different) and relate it to \(r(I)\). They show, by a class of examples that in some circumstances \(r(I) - \text{fp}(I)\) can be arbitrarily large. In a further example they prove that there exist ideals \(I\) in a one dimensional Cohen-Macaulay local domain \(R\) such that \(I\) is \(\mathcal M\)-primary and \(G(I)\) is Cohen-Macaulay, but \(F(I)\) is not Cohen-Macaulay. Moreover they prove that there exist local domains \(R\) of dimension greater than one such that the multiplicity of \(F(I)\) (as \(I\) varies over the ideals of \(R\) that have a principal reduction) cannot be bounded. Finally, they give a positive answer in several special cases to a question posed by \textit{S. Huckaba} [J. Algebra 108, 503-512 (1987; Zbl 0623.13011)].NEWLINENEWLINEFor the entire collection see [Zbl 0964.00058].