Wiseman–Wilson Theorem for Complex Conics

Definitions and mathematical statement

Let $V={\mathbb{C}}[X_0,X_1,X_2]_2$ be the six-dimensional vector space of homogeneous quadratic forms. For $0\ne q\in V$, write \[ C_q=\{x\in{\mathbb{P}}^2({\mathbb{C}}):q(x)=0\}. \] Forms differing by a nonzero scalar represent the same projective conic. For a finite subset $S\subset{\mathbb{P}}^2({\mathbb{C}})$, define \[ W(S)=\{q\in V:q(s)=0\text{ for every }s\in S\}. \] Here vanishing is independent of the homogeneous coordinates chosen for each point. Say that five distinct points $S$ uniquely determine a conic if $\dim_{{\mathbb{C}}}W(S)=1$. This uniqueness condition is stronger than merely lying on a conic.

Under the interpretation that conics include reducible and nonreduced quadratic curves, and that "determined" means uniquely determined among all projective conics, the supplied question is: \[ \begin{gathered} \text{For every finite }P\subset{\mathbb{P}}^2({\mathbb{C}})\text{ with }W(P)=\{0\},\\ \text{do there exist }S\subseteq P\text{ and }0\ne q\in V\text{ such that}\\ |S|=5,\qquad W(S)={\mathbb{C}} q,\qquad P\cap C_q=S\,? \end{gathered} \] The hypothesis says that no conic contains all of $P$; the conclusion asks for a uniquely determined conic containing exactly five points of $P$.

The supplied record does not explicitly specify whether reducible or nonreduced conics are admissible, or define "determined." Thus the displayed formulation is conditional on the stated conventions. If nonsingular conics are required, the conclusion must additionally require $C_q$ to be nonsingular; whether that restriction also changes the hypothesis requires clarification from the source. The supplied background also describes an incomplete higher-degree Sylvester--Gallai question, whereas the explicit statements concern degree two. No higher-degree assertion is reconstructed here.

Short English statement

The question asks whether every finite set of points in the complex projective plane that is not contained in a conic has a conic uniquely determined by five of its points and containing no others. The precise draft uses all quadratic conics; admissibility of singular conics and the intended uniqueness convention require review. The supplied record labels the problem open, without independent verification here.

Sources

• Supplied UnsolvedMath local registry record, permanent ID $30003084$, exact source problem code catalogue entry, titled "Wiseman--Wilson Theorem for Complex Conics," in the Oberwolfach Reports Open Problems collection. The record gives proposed year $2016$ and names no proposer.
• Mini-Workshop: Arrangements of Subvarieties, and their Applications in Algebraic Geometry (2016), Oberwolfach Reports, pp. 653--702, DOI 10.4171/owr/2016/14: https://doi.org/10.4171/owr/2016/14. This is the supplied report citation; no problem-specific page or passage is included.
• Supplied literature source labeled "Boys--Valculescu--De Zeeuw": https://doi.org/10.1016/j.ejc.2022.103509. The record supplies neither its title nor a specific result connecting it to this formulation.
• Literature assessment dated 2026-08-21: the supplied curation labels the problem open and describes the extracted lead-in as an incomplete higher-degree Sylvester--Gallai question, with no general theorem settling the intended statement beyond low-degree cases. This is supplied dated curation, not an independently verified literature assessment; its scope relative to the explicit complex-conic question requires review.
• The registry marks the statement corrected_verified and reports that the official report passage was checked and rewritten self-containedly. That verification claim is supplied metadata, not verification performed for this draft. The comparison with the real theorem is attributed to Wiseman--Wilson in the supplied statements; no original citation for that theorem is supplied.