Log abundance for threefolds in characteristic in numerical dimension one
The abundance conjecture predicts that if is a projective log canonical pair and is nef, then is semiample. For threefolds over this is a theorem (Miyaoka, Kawamata, Keel-Matsuki-McKernan). Over algebraically closed fields of characteristic it remains open; Zheng Xu proved nonvanishing for , abundance in nef dimension at most two, and abundance in numerical dimension two. Is abundance true for projective log canonical threefold pairs in characteristic , in particular when is nef of numerical dimension one?
- Result
- Proved(see note)
- Status
- Partial result
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Birational geometry in positive characteristic; minimal model program
- Posed by
- The abundance conjecture of the minimal model program (classical; the manuscript names no poser); the remaining numerical-dimension-one case is isolated by Zheng Xu's 2024 and 2026 papers
- Year posed
- —
- Years open
- —
- Solved
- 2026-10-05
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Unreviewed
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 18 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Principal Theorem 1.1: over an algebraically closed field of characteristic , if is a projective log canonical threefold pair with effective -boundary, is -Cartier, nef and of numerical dimension one, then is semiample; neither terminality nor -factoriality is needed. The companion proves the terminal -factorial case ( nef with gives and semiampleness). Characteristics 2 and 3, numerical dimension zero and higher dimensions are not addressed.
What the AI did
The release README says the results were produced by an unreleased internal OpenAI model with one fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region, whose write-up was human-edited). The manuscript is authored 'OpenAI' and names no human author. The principal manuscript (October 5) depends essentially on a companion of September 24 that proves the terminal case and an obstruction theorem for nef Cartier cycles.
Verification
No independent mathematician has checked this yet. Checked here: the introductions and main theorems of both manuscripts were read against the abundance conjecture. lean/docs/035.md does not exist at the pinned commit and formalization.yaml has no entry for either manuscript. The log result uses the companion's cycle obstruction as an essential input, so both papers must hold. Inputs from Xu's nonvanishing and numerical-dimension-two theorems and the positive-characteristic MMP are assumed.