Point Configurations Producing Unexpected Curves

Definitions and mathematical statement

The supplied question asks for a characterization of finite point configurations producing unexpected curves. It does not specify the ambient space, ground field, characteristic, curve degree, or fat-point multiplicity. The following gives a precise conditional plane-curve formulation; these additional parameters are not asserted to be hypotheses supplied by the source.

Suppose the intended ambient space is ${\mathbb{P}}^2_k$, where $k$ is a specified algebraically closed field. Let $Z\subset{\mathbb{P}}^2_k$ be a finite set of distinct points, regarded as a reduced subscheme, and let $\mathcal I_Z$ be its ideal sheaf. For integers $d\geq 0$ and $m\geq 1$, and a point $P\notin Z$, define the fat point $mP$ by the ideal sheaf $\mathcal I_P^m$. Set \[ V_d(Z)=H^0\left({\mathbb{P}}^2_k,\mathcal I_Z(d)\right),\qquad V_{d,m}(Z,P)=H^0\left({\mathbb{P}}^2_k,(\mathcal I_Z\cap\mathcal I_P^m)(d)\right). \] Thus $V_{d,m}(Z,P)$ consists of degree-$d$ homogeneous forms vanishing on $Z$ and to order at least $m$ at $P$. The restriction map \[ \rho_{Z,P}^{d,m}:V_d(Z)\longrightarrow H^0\left(mP,\mathcal O_{mP}(d)\right) \] has kernel $V_{d,m}(Z,P)$, and its target has dimension $\binom{m+1}{2}$. The expected vector-space dimension is therefore \[ e_{d,m}(Z)=\max\left\{0,\dim_k V_d(Z)-\binom{m+1}{2}\right\}. \] For these specified parameters, define the unexpectedness condition by \[ \dim_k V_{d,m}(Z,P)>e_{d,m}(Z) \quad\text{for general }P\in{\mathbb{P}}^2_k\setminus Z. \] Here "general" means that the inequality holds for every $P$ in some nonempty Zariski-open subset of ${\mathbb{P}}^2_k\setminus Z$. Equivalently, the restriction map fails to have maximal rank for general $P$.

Requested characterization, subject to unresolved scope. Characterize the finite point sets $Z$ satisfying this unexpectedness condition in the ambient setting and degree--multiplicity regime intended by the source. The supplied record does not determine whether $d,m$ are fixed, whether existence of a pair $(d,m)$ is intended, or whether a prescribed relation between them is required. It also does not establish that the conditional plane-curve setting above is the complete intended setting. Consequently, a unique fully specified classification problem cannot be recovered from the supplied wording alone.

Short English statement

Characterize finite point configurations for which requiring a curve through the configuration to have a specified multiplicity at a general additional point leaves more curves than the expected dimension predicts. A precise plane-curve interpretation is given conditionally; the source leaves the ground field, ambient space, and degree--multiplicity quantifiers unspecified.

Sources

• Local UnsolvedMath registry, supplied complete record: permanent ID $30003075$, title Point Configurations Producing Unexpected Curves, exact source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record gives the proposed year as 2016 and supplies no individual proposer.
• Supplied primary provenance: Mini-Workshop: Arrangements of Subvarieties, and their Applications in Algebraic Geometry (2016), Oberwolfach Reports, pp. 653--702, DOI 10.4171/owr/2016/14. Supplied source URL and literature-source label "OWR report 14/2016": https://doi.org/10.4171/owr/2016/14.
• Supplied literature source: Cook--Harbourne--Migliore--Nagel, https://doi.org/10.1112/S0010437X18007376. The supplied record does not give a title or a specific theorem reference.
• Supplied literature source: Dimca--Sticlaru, https://doi.org/10.1007/s10711-021-00602-5. The supplied record does not give a title or a specific theorem reference.
• Literature assessment date: 2026-08-21. This is supplied dated curation, not independently verified. The supplied status is "partially solved". Its assessment says that the broad classification question remains open, that an unexpected-curve criterion is characterized through the splitting type of the dual line arrangement, and that many explicit families and low-degree cases are known. The background describes this as dated literature triage derived from the supplied Oberwolfach curation and calls for independent verification of time-sensitive status and cited claims.
• The supplied statement-status label is corrected_verified; its accompanying qualification says only that notation, typography, and sentence structure were repaired without changing mathematical content. This label is not treated here as independent mathematical verification.