Algebraic geometer Jenia Tevelev writes out OpenAI’s ‘remarkably simple’ proof and calls the release ‘a stunning achievement’
Result confirmed
Importance: historic (5 of 5)The takeaway
Nagata’s conjecture on plane curves through very general points, open since 1959, was one of the claims in OpenAI’s 6 Oct maths release. A UMass Amherst specialist has now reorganized, simplified and published the full proof himself, while criticizing how the manuscripts were dumped on the community.
Status
- Claim
Result confirmed
- Our reporting
- Medium confidence
- Verification
- Preprint only
- Importance
- Historic (5 of 5)
- Last verified
- 22 October 2026
Your AI and this story
- GPT-6 Astra174 days after its cutoff
- Claude Opus 5.5113 days after its cutoff
- Gemini 4 Argon204 days after its cutoff
- Grok 4.7143 days after its cutoff
None of these four assistants can know about it. The closest, Claude Opus 5.5, stops 113 days before it.
Key facts
- Problem: Nagata (Amer. J. Math. 81, 1959), arising from his counterexample to Hilbert’s 14th problem; Nagata proved it only when r is a perfect square
- Claim: OpenAI family 039, ‘Nagata’s conjecture and maximal Seshadri constants’: strict inequality Σ m_i < d√r for every nonzero effective plane curve through r ≥ 10 very general complex points, plus maximal multipoint Seshadri constants (L^n/r)^(1/n) for every smooth polarized variety and all large r
- Exposition: Jenia Tevelev, ‘On OpenAI’s proof of the Nagata conjecture’, arXiv 2610.30050 [math.AG], 21 Oct 2026; expanded account of his talk at UMass Amherst
- Method (abstract): reduction to the normal cone and collision of fat points into one monomial scheme reduce the problem to interpolation on the normal ruled surface of a smooth plane cubic; ‘the novelty is that the study of the asymptotics of theta functions as the cubic develops a node, together with interpolation duality, reduces the problem further to interpolation on a triangle of lattice points, which is straightforward’
- Tevelev: ‘I have reorganized and simplified the argument by working directly on the normal ruled surface of a smooth plane cubic’; he follows the complex-geometry manuscript and has ‘not studied in detail’ the algebraic companion manuscripts on Seshadri constants
- Tevelev: ‘At the same time, I think these manuscripts are a stunning achievement. The model managed to connect the dots and solve the problems that many of the experts thought were out of reach. I think a wholesale re-evaluation of the mathematical corpus is currently underway, with the old underbrush of mathematical conjectures rapidly being burned in the AI fire.’
- Criticism: ‘As I read the released manuscripts, I wondered whether I might be their first human reader. They seemed almost entirely unedited.’ He proposes that AI companies fund an independent, NSF-style foundation for mathematics research
- AI disclosure (Tevelev’s notes): ‘These notes were edited and illustrated with ChatGPT Astra.’
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- Extra result in the notes: the same calculation determines the nef and pseudoeffective cones of a weighted blow-up of P² at one point for very general local parameters when the ratio of weights exceeds nine
What happened
OpenAI’s 6 Oct release listed, as family 039, a proof of Nagata’s conjecture in its strict, nonhomogeneous form, together with three companion manuscripts on maximal multipoint Seshadri constants (surfaces, higher dimensions, positive characteristic). According to the release, the manuscripts were written by an OpenAI model with no human author.
On 21 Oct Jenia Tevelev (UMass Amherst) posted arXiv 2610.300501, notes expanding a talk he gave at UMass Amherst. He claims no originality: he reorganizes the argument, works directly on the normal ruled surface of a smooth plane cubic, and proves the conjecture in full (Theorem A). He also derives the nef and pseudoeffective cones of certain weighted blow-ups of P².
The notes end with a commentary. Tevelev calls the manuscripts “a stunning achievement” and says that “the old underbrush of mathematical conjectures” is “rapidly being burned in the AI fire”. He also says the release, apparently unedited, leaves the community a large unpaid editorial job that “harms our profession, especially the motivation and training of young people”, and he proposes an independent research foundation funded by AI companies.
No other expert reactions were found on 22 Oct (one web search; X not searched).
Why it matters
Nagata’s conjecture is one of the best-known open problems in the geometry of plane curves. With this note it joins Thompson’s group F among the 6 Oct claims that specialists have reconstructed by hand. The note’s closing section is also an unusually direct statement, from a working expert, of both the achievement and the cost to the field.
Sources
3 sources from 2 sites. Numbers match the chips in the text.
3 sources: 3 primary
Primary
- arXiv 2610.30050: Tevelev, On OpenAI’s proof of the Nagata conjecturearxiv.org, paper
- OpenAI: Nagata’s conjecture for plane curves (23 Sep manuscript)github.com, paper
- openai/math CONTENTS.md (family 039)github.com, code
Changes
- Filed (PDF read in full text)