On the stable Harbourne conjecture for ideals defining space monomial curves

Abstract

For the ideal $\mathfrak{p}$ in $k[x,y,z]$ defining a space monomial curve, we show that ${\mathfrak{p}}^{(2n-1)}\subseteq \mathfrak{m}{\mathfrak{p}}^n$ for some positive integer $n$, where $\mathfrak{m}$ is the maximal ideal $(x,y,z)$. Moreover, the smallest such $n$ is determined. It turns out that there is a counterexample to a claim due to Grifo, Huneke, and Mukundan, which states that ${\mathfrak{p}}^{(3)}\subseteq \mathfrak{m}{\mathfrak{p}}^2$ if $k$ is a field of characteristic not $3$; however, the stable Harbourne conjecture holds for space monomial curves as they claimed.