Cohen-Macaulayness and negativity of \(A\)-invariants in Rees algebras associated to \(\mathfrak m\)-primary ideals of minimal multiplicity (Q1582736)

Let \((A,{\mathfrak m})\) be a CM local ring with infinite residue field, \(I\) an \({\mathfrak m}\)-primary ideal with minimal multiplicity. The first and main aim of the paper is to give characterizations for the Cohen-Macaulayness and Gorensteinness of the Rees algebra \(R(I)\) of \(I\). Then, if \(\dim A=3\) and the Cohen-Macaulay type of \(A\), \(r(A)\), is not bigger than 3 it is shown that \(R(I)\) is Cohen-Macaulay iff \(\text{Proj}(I)\) is a Cohen-Macaulay scheme and the \(i\)-th \(a\)-invariant of \(R(I)\) is negative for all \(i\in \mathbb{Z}\). Then, an example shows that the hypothesis that \(r(A)\leq 3\) is necessary, giving thus a counter-example to a conjecture of \textit{T. Korb} and \textit{Y. Nakamura} [J. Math. Soc. Japan 50, No. 2, 451-467 (1998; Zbl 0902.13008)].