Containment of the Symbolic Cube in the Square for the Roulleau–Urzúa Configurations of Cubics

In an Oberwolfach report of 2015 Harbourne considered the configurations \(C_n\) of \(\frac43(n^2-3)\) plane cubics due to Roulleau and Urzúa, defined when \(3\) divides \(n\), and asked whether \(I^{(3)}\not\subseteq I^2\) for the ideal \(I\) of the singular points of \(C_n\). He reported that for \(C_3\) the containment \(I^{(3)}\subseteq I^2\) holds, with \(I\) the ideal of the \(12\) proper singular points taken with multiplicity \(2\) and of the \(24\) infinitely near triple points taken with multiplicity \(1\). For \(n=3m\) we take \(I\) to be the ideal of the cluster of all singular points of \(C_n\), proper and infinitely near, with weight \(3m^2-1\) at each of the \(12\) points of multiplicity \(n^2-3\) and weight \(1\) at each infinitely near triple point and at each quadruple point; for \(m=1\) this is Harbourne's ideal, and for \(m\ge2\) it is our extension of his description. We prove that \(I^{(3)}\subseteq I^2\) for every \(m\ge1\), so that the answer to the question is negative for every \(n\) divisible by \(3\). From the description of \(C_n\) as the union of the Halphen cubics of order \(m\), due to Bauer, Harbourne, Roé and Szemberg, Bézout's theorem gives \(\alpha(I^{(3)})=36m^2-12\), and for \(m\ge2\) the Kawamata–Viehweg vanishing theorem on the blow-up of the cluster gives \(\operatorname{reg}(I)\le18m^2-8\); for \(m=1\) an exact computation, made with a computer, gives \(\operatorname{reg}(I)=12\), the value reported by Harbourne. The containment then follows from a criterion of Bocci and Harbourne. The question, the case \(n=3\) and its numbers are Harbourne's; for \(m\ge2\) we found no answer in the literature accessible to us. The second part of the question, on interesting analogous configurations of conics, is not addressed. This is an unrefereed note.