ACM Line Configurations and Symbolic Initial Degrees

Definitions and mathematical statement

The corrected target supplied for this record is the classification of arithmetically Cohen--Macaulay configurations of lines in projective three-space. The source does not specify a ground field, its characteristic, or whether it is algebraically closed. Accordingly, the formulation below is relative to a field $k$; the intended restrictions on $k$ remain unresolved.

Under the interpretation of a line configuration as a nonempty finite reduced union of distinct projective lines, let \[ S=k[x_0,x_1,x_2,x_3],\qquad X=\bigcup_{i=1}^{r}L_i\subseteq\mathbb P^3_k,\qquad r\geq 1, \] where each $L_i$ is a projective linear subspace of dimension one defined over $k$. Give $X$ its reduced scheme structure, and set \[ I(X)=\bigcap_{i=1}^{r}I(L_i),\qquad A_X=S/I(X). \] The configuration is arithmetically Cohen--Macaulay (ACM) if its homogeneous coordinate ring $A_X$ is Cohen--Macaulay. Since $\dim A_X=2$, this is equivalent to \[ \operatorname{depth}_{S_+}A_X=2, \qquad S_+=(x_0,x_1,x_2,x_3). \] Classification problem. Characterize all such configurations $\{L_1,\ldots,L_r\}$, for arbitrary positive integers $r$, for which $A_X$ is Cohen--Macaulay. The finite reduced-union interpretation and the intended ground-field hypotheses require confirmation from the source; no additional restrictions on intersections or positions of the lines are supplied.

Source qualification. The original registry wording also asked which reduced, possibly irreducible, curves $C\subseteq\mathbb P^3$ satisfy \[ \alpha\left(I(C)^{(2)}\right)-\alpha\left(I(C)\right)=1. \] Here $\alpha(J)=\min\{d\geq 0:J_d\neq 0\}$ is the initial degree of a nonzero homogeneous ideal, and, for a radical homogeneous ideal $I$, its second symbolic power is \[ I^{(2)}=\bigcap_{\mathfrak p\in\operatorname{Min}(I)}\left(I^2S_{\mathfrak p}\cap S\right). \] The supplied statement-verification note assigns that curve question to the separate source code OWR-13494-013; it is not an additional condition on the configurations in this record. The supplied literature assessment discusses an ACM subcase and the unresolved existence of a non-ACM line configuration with this initial-degree gap. Its applicability to the unrestricted ACM classification problem above is unclear.

Short English statement

Characterize the finite reduced unions of lines in $\mathbb P^3$ whose homogeneous coordinate rings are Cohen--Macaulay. The intended ground field and precise convention for a line configuration need confirmation. The supplied correction separates the symbolic-initial-degree question for reduced curves into OWR-13494-013.

Sources

• Local UnsolvedMath registry, supplied complete record with permanent ID $30002793$, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The supplied proposer field is empty; the proposed year is 2015.
• 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 and literature-source link: https://doi.org/10.4171/owr/2015/9.
• Janssen, On the fattening of lines in P3. Supplied literature-source link: https://doi.org/10.1016/j.jpaa.2014.05.033. No theorem number or theorem statement is supplied.
• The supplied literature assessment is dated 2026-08-21. It labels the problem open and states that an ACM subcase is classified by a cited theorem, while the existence of a non-ACM line configuration with the stated initial-degree gap remains unresolved. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment; its relation to the corrected classification target requires review.
• The registry labels the statement corrected_verified and reports that the official report separates the ACM classification question from the curve symbolic-initial-degree question, assigning the latter to OWR-13494-013. These are supplied verification claims; the linked sources have not been independently consulted for this draft.