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

Abstract

In an Oberwolfach report of 2015 Harbourne considered the configurations C_n of 4(n² − 3)/3 plane cubics due to Roulleau and Urzúa, defined when 3 divides n, and asked whether I^(3) ⊄ I² for the ideal I of the singular points of C_n. He reported that for C_3 the containment I^(3) ⊆ I² 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 3m² − 1 at each of the 12 points of multiplicity n² − 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 ≥ 2 it is our extension of his description. We prove that I^(3) ⊆ I² for every m ≥ 1, 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 α(I^(3)) = 36m² − 12, and for m ≥ 2 the Kawamata–Viehweg vanishing theorem on the blow-up of the cluster gives reg(I) ≤ 18m² − 8; for m = 1 an exact computation, made with a computer, gives 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 ≥ 2 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. Unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text. Corpus identifier: OWR-13494-009 (Harbourne, H-constants and the containment problem, Oberwolfach Report 9/2015, Example 3).