On the stable Harbourne conjecture for ideals defining space monomial curves

TL;DR

This paper proves the stable Harbourne conjecture for ideals defining space monomial curves, establishing specific containments between symbolic and ordinary powers, and clarifies the minimal such power, while also providing a counterexample to a previous claim.

Contribution

It demonstrates the stable Harbourne conjecture for space monomial curves and determines the minimal integer for the containment, correcting a prior misconception.

Findings

Proved the containment e(p)^{(2n-1)}  e(m)  e(p)^n for some n.

Determined the smallest such n for the containment.

Provided a counterexample to a previous claim about symbolic powers.

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.