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 $\mathfrak {p}^{(2 n - 1)} \subseteq \mathfrak {m} \mathfrak {p}^{n}$ 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.