Symbolic Containment for Roulleau–Urzúa Configurations

Definitions and mathematical statement

Let $C_n$ denote the configurations of cubic curves attributed in the supplied record to Roulleau--Urzúa, and let $Z_n$ denote their intended sets of singular points. The record does not supply the construction of $C_n$, the admissible indices $n$, the ground field, the ambient space, or the precise convention for selecting singular points. These data must be recovered from the cited source before the question becomes fully specified.

The containment can be formulated conditionally as follows. Once the ambient homogeneous coordinate ring $R$ is specified, interpret the ideal of the singular points as the radical homogeneous ideal $I_n=I(Z_n)\subseteq R$. For a radical ideal $I$ in a Noetherian ring $R$, write \[ I^{(3)}=\bigcap_{\mathfrak p\in\operatorname{Min}(I)}\left(I^3R_{\mathfrak p}\cap R\right), \qquad I^2=\left\{\sum_{j=1}^{r}f_jg_j:r\geq 0,\ f_j,g_j\in I\right\}, \] where $\operatorname{Min}(I)$ is the set of prime ideals minimal over $I$, and the intersection with $R$ denotes contraction from the localization. Thus $I^{(3)}$ is the third symbolic power and $I^2$ is the ordinary square. The reduced-point interpretation is explicit here; the supplied record does not specify a scheme structure.

The first question is whether \[ \forall n\text{ for which the intended }C_n\text{ is defined}, \qquad I_n^{(3)}\subseteq I_n^2. \] The second question asks for interesting analogous configurations of conics. The record does not define the required analogy, prescribe incidence or singularity conditions, or specify whether such conic configurations should satisfy or violate the corresponding containment. This second request therefore remains an open-ended construction question rather than a uniquely specified mathematical assertion.

Short English statement

For every admissible Roulleau--Urzúa cubic configuration, must the third symbolic power of its singular-point ideal lie in its ordinary square? The source also asks for analogous conic configurations. The family, ambient setting, admissible indices, and intended conic analogy require clarification from the cited source.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30002790$, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record attributes the configurations to Roulleau--Urzúa, gives the proposed year as 2015, and names no individual proposer.
• Mini-Workshop: Ideals of Linear Subspaces, Their Symbolic Powers and Waring Problems (2015), Oberwolfach Reports, pp. 489--532, DOI 10.4171/owr/2015/9. Supplied report reference: https://doi.org/10.4171/owr/2015/9.
• Bauer, Harbourne, Roe, and Szemberg, Halphen cubics, supplied literature reference: https://doi.org/10.1007/s13348-016-0172-4.
• Supplied literature assessment dated exactly 2026-08-21: the record labels the problem open, describes the $C_3$ member as verified in the record, and reports that no authoritative result was found settling the symbolic-cube containment for the full family or the analogous conic question. Its listed checked sources are Halphen cubics and the original OWR report linked above. This is supplied dated curation, explicitly described as literature triage requiring independent verification; neither the assessment nor the cited claims have been independently verified here.
• The registry labels the statement corrected_verified and says its mathematical content was preserved while notation, typography, or omitted definitions were repaired for publication. This supplied editorial label does not establish mathematical correctness or resolution of the problem.