The Dolgachev-Weisfeiler affine-fibration conjecture
An -fibration is a morphism whose fiber over every point is affine -space over the residue field. Dolgachev and Weisfeiler (1974, Section 3.8.5) predicted that an affine morphism to a normal, locally noetherian, integral base all of whose fibers are over their residue fields is Zariski-locally trivial, that is, locally a product with . Is every such affine-space fibration over a normal integral base, for instance over or over , a Zariski-locally trivial -bundle?
- Result
- Disproved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Construction
- Field
- Affine algebraic geometry, affine fibrations
- Posed by
- B. Yu. Weisfeiler and I. V. Dolgachev, Unipotent group schemes over integral rings, Math. USSR-Izv. 8 (1974), Section 3.8.5
- Year posed
- 1974
- Years open
- 52y
- Solved
- 2026-09-23
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Unreviewed
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 35 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Corollary 7.1: for the fourfold of the cancellation counterexample, the maps (given by ) and are smooth surjective morphisms of relative dimension three whose fiber over every point is over the residue field, yet neither is Zariski-locally trivial. Local triviality would make a locally polynomial algebra over and hence, by Bass-Connell-Wright and Quillen-Suslin, a polynomial ring, contradicting Theorem 1.1. So the conjecture fails in relative dimension three over the smooth bases and . Relative dimension two is not addressed.
What the AI did
Produced by an unreleased internal OpenAI model as part of an OpenAI evaluation on open research problems. The release README says the vast majority of results used one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result; this result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is authored as OpenAI with no human author named. The README also cautions that unformalized results could have issues.
Verification
No independent mathematician has checked this yet. Checked here: the abstract, the introduction's paragraph on affine-space fibrations and Corollary 7.1 with its proof outline, read against the conjecture as the manuscript cites it (Dolgachev-Weisfeiler 1974, Section 3.8.5). The deduction itself is short given Theorem 1.1. Not Lean-checked as stated: the release formalizes the non-polynomiality and stabilization of A (Comparator challenge ComplexCancellation, which the deduction rests on), but its scope note says the further consequences are not included, and the fibration statement has no Lean counterpart. The formal check covers the key input only.