Baker's question asks whether a fixed smooth variety 
over a finite
field, which is a quasiprojective variety
of pure positive dimension, has the following property.
For every sufficiently large 
, every embedding 
into projective space
whose connected components lie in no hyperplane
admits a hyperplane defined over the field whose intersection
with 
is a smooth variety of dimension 
(Poonen 2004, Question 4.1).
The question is related to Bertini's theorem, but varies the embedding and keeps the polynomial degree of the section equal to 1. Poonen's finite-field Bertini theorem instead fixes the embedding and allows the polynomial degree of a hypersurface to grow (Poonen 2004).
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).
