Non-ACM Line Configurations with Minimal Symbolic Initial-Degree Gap

Definitions and mathematical statement

Let $k$ be a field and let $S=k[x_0,x_1,x_2,x_3]$ have its standard grading. Under the usual interpretation of a line configuration as a nonempty finite reduced union of distinct projective lines, write \[ L=\ell_1\cup\cdots\cup\ell_r\subseteq\mathbb P^3_k,\qquad r\geq1, \] and let $\mathfrak p_i\subset S$ be the homogeneous prime ideal of $\ell_i$. Its saturated homogeneous ideal is \[ I=I(L)=\bigcap_{i=1}^r\mathfrak p_i. \] For a nonzero homogeneous ideal $J\subseteq S$, define its initial degree by \[ \alpha(J)=\min\{d\in\mathbb Z_{\geq0}:J_d\neq0\}. \] The second symbolic power of this reduced line ideal is \[ I^{(2)}=\bigcap_{i=1}^r\left(I^2S_{\mathfrak p_i}\cap S\right) =\bigcap_{i=1}^r\mathfrak p_i^2. \] The configuration is arithmetically Cohen--Macaulay (ACM) when $S/I$ is Cohen--Macaulay; here $\dim(S/I)=2$, so this means $\operatorname{depth}_{(x_0,x_1,x_2,x_3)}(S/I)=2$.

The supplied question is whether there exists such a configuration $L$ for which $S/I$ is not Cohen--Macaulay and \[ \alpha\left(I^{(2)}\right)-\alpha(I)=1. \] The source record does not specify the ground field, its characteristic, or whether it is algebraically closed. Thus the intended choice of $k$, and any quantifier over ground fields, remain unresolved. The finite reduced interpretation of "configuration of lines" used above also requires confirmation against the original report; it is an explicit interpretation, not an additional supplied hypothesis.

Short English statement

The question asks whether a non-ACM union of projective lines in $\mathbb P^3$ can have its second symbolic power begin exactly one degree above its defining ideal. The ground field and the precise convention for a line configuration are not specified in the supplied record.

Sources

• Local UnsolvedMath registry, permanent ID $30002792$, Oberwolfach Reports Open Problems collection, source problem code catalogue entry; proposed year recorded as 2015. No individual proposer is supplied.
• Mini-Workshop: Ideals of Linear Subspaces, Their Symbolic Powers and Waring Problems (2015), Oberwolfach Reports, pp. 489--532, DOI: https://doi.org/10.4171/owr/2015/9. The supplied background describes these reports as records of mathematical workshops and their open-problem discussions.
• Supplied literature reference: "Janssen, fattening of lines in P3," https://doi.org/10.1016/j.jpaa.2014.05.033.
• Supplied literature reference: "original OWR report," https://doi.org/10.4171/owr/2015/9.
• Literature assessment checked at 2026-08-21: the supplied curation labels the problem open and reports that the ACM hypothesis yields a classification involving pseudostars and coplanar configurations, while no result was found deciding the stated existence question for non-ACM line configurations. This is supplied dated literature triage derived from the OWR curation, not an independently verified assessment; its status and cited claims require independent verification.
• The registry supplies the statement-status label corrected_verified, with the explanation that mathematical content was preserved while notation, typography, or omitted definitions were repaired for publication. This label is source metadata, not independent mathematical verification of this draft.