Complex Line Arrangements without Double Points

Definitions and mathematical statement

Under the usual convention that a line arrangement is a finite set of distinct lines, let $\mathcal A=\{L_1,\ldots,L_d\}$ be a complex line arrangement in an ambient plane $X$. The supplied record does not specify whether $X={\mathbb{P}}^2({\mathbb{C}})$ or $X={\mathbb{C}}^2$; the definitions below apply after that choice is fixed. Define the intersection set and the multiplicity of an intersection point by \[ S(\mathcal A)=\bigcup_{1\le i<j\le d}(L_i\cap L_j),\qquad m_{\mathcal A}(p)=\#\{i:p\in L_i\}\quad(p\in S(\mathcal A)). \] For each integer $r\ge2$, put \[ t_r(\mathcal A)=\#\{p\in S(\mathcal A):m_{\mathcal A}(p)=r\}. \] An ordinary double point is an intersection point of multiplicity exactly two.

Classification problem. Classify complex line arrangements $\mathcal A$ satisfying \[ t_2(\mathcal A)=0, \] equivalently, satisfying \[ \forall p\in S(\mathcal A),\qquad m_{\mathcal A}(p)\ge3. \] The condition permits intersection points of multiplicity greater than two. No lower bound on $d$, nonconcurrency hypothesis, or exclusion of arrangements with no intersection points is supplied. The intended ambient plane and the equivalence relation for the classification are unresolved: the record does not say whether the target is a classification of incidence structures or of geometric realizations up to a specified transformation group.

Short English statement

Classify finite arrangements of distinct complex lines for which every intersection point lies on at least three lines. The supplied record leaves the affine or projective setting and the meaning of equivalence in the classification unspecified.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004606$, title Complex Line Arrangements without Double Points, exact source problem code catalogue entry; collection: Oberwolfach Reports Open Problems. The record gives proposed year 2021 and names no proposer.
• Supplied source citation: Logarithmic Vector Fields and Freeness of Divisors and Arrangements: New perspectives and applications (2022), Oberwolfach Reports, pp. 225--300, DOI 10.4171/owr/2021/5. https://doi.org/10.4171/owr/2021/5. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions; the literature list labels this reference Oberwolfach Report 5/2021.
• Supplied literature reference: Hanumanthu--Harbourne, Journal of Algebraic Combinatorics. https://doi.org/10.1007/s10801-020-00987-8. No article title or further bibliographic details are supplied.
• Supplied literature assessment dated 2026-08-22: the classification remains open, and the literature identifies several known types rather than a complete list. This is supplied dated curation derived from the OWR record, not an independently verified assessment. The source expressly requests independent verification of time-sensitive status and cited claims.