Curves with Minimal Symbolic Initial-Degree Gap

Definitions and mathematical statement

Let $k$ be a field and let $S=k[x_0,x_1,x_2,x_3]$ with its standard grading. The supplied record does not specify the intended base field, its characteristic, or whether it must be algebraically closed; the formulation below is conditional on these unresolved choices.

For a nonempty reduced projective curve $C\subseteq{\mathbb{P}}^3_k$, let $I=I(C)\subset S$ denote its saturated homogeneous vanishing ideal. Here the conventional interpretation of a curve as a scheme of pure dimension one is used; the supplied record does not explicitly settle this convention. Neither irreducibility nor reducibility is required. Define the initial degree of a nonzero homogeneous ideal $J\subset S$ by \[ \alpha(J)=\min\{d\in{\mathbb{Z}}_{\geq0}:J_d\neq0\}. \] For the radical ideal $I$, define its second symbolic power by \[ I^{(2)}=\bigcap_{\mathfrak p\in\operatorname{Min}(I)}\left(I^2S_{\mathfrak p}\cap S\right), \] where $\operatorname{Min}(I)$ is the set of minimal prime ideals over $I$, and each intersection with $S$ denotes contraction from the localization.

Classification problem. Determine all reduced curves $C\subseteq{\mathbb{P}}^3_k$, allowing irreducible curves, such that \[ \alpha\left(I(C)^{(2)}\right)-\alpha\left(I(C)\right)=1. \] No restriction to line arrangements or arithmetically Cohen--Macaulay curves is stated in the supplied problem. A definitive formulation requires clarification of the base-field hypotheses and the convention for curves.

Short English statement

Classify the reduced projective curves in three-dimensional projective space whose second symbolic power has initial degree exactly one greater than that of the curve's ideal. Irreducible curves are allowed. The intended base field and the precise convention for a curve remain unspecified.

Sources

• Local UnsolvedMath registry, permanent ID $30002794$, title Curves with Minimal Symbolic Initial-Degree Gap, Oberwolfach Reports Open Problems collection; exact source code catalogue entry. The supplied record 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: https://doi.org/10.4171/owr/2015/9. The supplied background describes these reports as documenting mathematical workshops and their open-problem discussions. This reference is also supplied as a literature source.
• Janssen, On the fattening of lines in P3, supplied literature source: https://doi.org/10.1016/j.jpaa.2014.05.033.
• Supplied literature assessment, checked 2026-08-21: the registry labels the problem open and states that results cover arithmetically Cohen--Macaulay (ACM) arrangements and special configurations, but that no complete classification of all reduced curves in ${\mathbb{P}}^3$ with this symbolic-degree gap is established. This is supplied dated curation derived from the OWR record, not an independently verified assessment. The supplied background explicitly requests independent verification of time-sensitive status and cited claims.