Bertini's theorem states that the general curve of a system which is linearly dependent on a certain number of given irreducible algebraic curves will not have a singular point which is not fixed for all the curves of the system.
Over a finite field, a nonempty open set need not contain a rational point. Poonen (2004) proved a finite-field version in which the polynomial degree of the intersecting hypersurface grows while the embedding of a smooth variety is fixed. Baker's question (Poonen 2004, Question 4.1) instead asks whether every sufficiently high-dimensional embedding of a fixed quasiprojective variety that is smooth has a smooth hyperplane section, provided no connected component lies in a hyperplane.
Zhang and Yang (2026a) reported a negative answer for every nonempty smooth variety 
over a finite field 
whose components all have the same positive dimension
and which is a quasiprojective variety.
For every sufficiently large 
, they construct a locally closed embedding 
into projective
space such that no connected component
lies in a hyperplane, even after any field
extension, but every hyperplane defined over 
cuts out a section with a singular
point. The claim concerns particular embeddings, not all embeddings.
The authors used GPT-based models for proof strategies and Rethlas for advisory proof checking. As of Oct. 2, 2026, the accompanying Lean development remained conditional on geometric lemmas left unproved in Lean, and independent specialist review of the complete argument had not been reported (Zhang and Yang 2026b, VibeMathed 2026).
