On the stable Harbourne conjecture for ideals defining space monomial curves

Abstract

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