On the equality of ordinary and symbolic powers of ideals
Let \(R\) be a Noetherian regular local ring of dimension \(d\) and let \(P\) be a prime ideal of height \(d-1\) in \(R\). For all \(n \in \mathbb N\) let \(P^{(n)} = P^nR_P \cap R\) denote the \(n\)-th symbolic power of \(P\). Craig Huneke asked the following question: ``If \(P^n = P^{(n)}\) for all \(n \leq d-1\), then is \(P^n = P^{(n)}\) for all \(n\)?''. He gave positive answers in embedding dimension \(d=3\), and embedding dimension \(d= 4\) provided \(R/P\) is Gorenstein. Furthermore, there are supporting examples showing that the Gorenstein property of \(R/P\) may already be forced by the equality \(P^2 = P^{(2)}\). Thus it is natural to ask: ``If \(P^2 = P^{(2)}\), then is \(R/P\) Gorenstein?'' In this article, the authors study these questions in the case of a monomial curve \(k[[t^{a_1},\ldots,t^{a_d}]]\). They provide validity evidence in some subcases of \(d=4\), and for any embedding dimension \(d\) either assuming that the numerical sequence \(a_1,\ldots,a_n\) contains an arithmetical subsequence of length \(5\), or assuming some upper bounds on the Hilbert-Samuel multiplicity of \(R/P\). Finally, using a deformation argument due to \textit{M. Morales} [J. Algebra 140, No. 1, 12--25 (1991; Zbl 0738.13012)], they extend their results to successive modifications of the originally considered monomial curve.
scientific article