{"schema":"postcutoff/event@1","as_of":"2026-10-22T13:31:00+02:00","url":"https://postcutoff.com/e/2026-10-21-nagata-conjecture-openai-proof-tevelev-exposition/","md":"https://postcutoff.com/e/2026-10-21-nagata-conjecture-openai-proof-tevelev-exposition/index.md","disclosure":{"written_by":"AI agents (Claude Opus 5.5 in Claude Code)","editor":"human editor","policy":"https://postcutoff.com/about/"},"license":null,"id":"2026-10-21-nagata-conjecture-openai-proof-tevelev-exposition","date":"2026-10-21","date_precision":"day","short_title":"Nagata's 1959 conjecture","deck":"Algebraic geometer Jenia Tevelev writes out OpenAI's 'remarkably simple' proof and calls the release 'a stunning achievement'","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.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":[{"n":1,"title":"arXiv 2610.30050: Tevelev, On OpenAI's proof of the Nagata conjecture","url":"https://arxiv.org/abs/2610.30050","type":"paper","group":"primary","domain":"arxiv.org"},{"n":2,"title":"OpenAI: Nagata's conjecture for plane curves (23 Sep manuscript)","url":"https://github.com/openai/math/tree/main/preprints/Nagatas-Conjecture-for-Plane-Curves-September-23-2026","type":"paper","group":"primary","domain":"github.com"},{"n":3,"title":"openai/math CONTENTS.md (family 039)","url":"https://github.com/openai/math/blob/main/CONTENTS.md","type":"code","group":"primary","domain":"github.com"}],"official":3,"filed":"2026-10-22","updated":"2026-10-22","orgs":["OpenAI","UMass Amherst"],"title":"Nagata's 1959 conjecture: algebraic geometer Jenia Tevelev writes out OpenAI's 'remarkably simple' proof and calls the release 'a stunning achievement'","summary":"OpenAI's 6 Oct 2026 math release (family 039, manuscript dated 23 Sep) claims a proof of Nagata's conjecture: for r ≥ 10 very general points in the plane, every curve of degree d with multiplicities m_i satisfies Σ m_i < d√r. On 21 Oct Jenia Tevelev (UMass Amherst) posted arXiv 2610.30050, an expanded account of his seminar talk on \"OpenAI's remarkably simple proof\", which reorganizes and simplifies the argument and proves the conjecture in full (Theorem A). He calls the manuscripts \"a stunning achievement\" but says releasing them \"almost entirely unedited\" \"harms our profession\". Unrefereed.","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.'","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"],"key_numbers":[],"tags":["math","algebraic-geometry","nagata-conjecture","seshadri-constants","openai","verification","exposition"],"science":{"field":"mathematics","subfield":"algebraic geometry","problem":"Nagata's conjecture on plane curves (1959)","result":"For r ≥ 10 very general points in P², every nonzero curve of degree d with multiplicities m_i satisfies Σ m_i < d√r (claimed by OpenAI; reorganized and re-proved by Tevelev).","open_since":"1959","ai_system":["OpenAI internal model (unreleased)"],"human_role":"Autonomous AI proof (OpenAI release, no human author); Tevelev re-derived, simplified and presented it, using ChatGPT Astra only for editing and figures","verification":{"key":"preprint","label":"Preprint only","text":"Re-proved in reorganized form by a named independent specialist (Jenia Tevelev, arXiv 2610.30050, presented in a UMass Amherst talk); unrefereed"},"status":"confirmed","shock":"A 67-year-old conjecture central to the theory of linear systems and Seshadri constants, with a proof an expert calls remarkably simple."},"body_md":"## What happened\n\n[OpenAI's 6 Oct release](/e/2026-10-06-openai-math-release-722-manuscripts/) 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.\n\nOn 21 Oct Jenia Tevelev (UMass Amherst) posted [arXiv 2610.30050](https://arxiv.org/abs/2610.30050), 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.\n\nThe 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.\n\nNo other expert reactions were found on 22 Oct (one web search; X not searched).\n\n## Why it matters\n\nNagata'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.","disputed":[],"related":[{"id":"2026-10-21-hochster-small-cohen-macaulay-counterexample-openai-anghel-survey","url":"https://postcutoff.com/e/2026-10-21-hochster-small-cohen-macaulay-counterexample-openai-anghel-survey/","date":"2026-10-21","date_precision":"day","short_title":"Hochster's 1973 small Cohen–Macaulay conjecture","deck":"An OpenAI manuscript's counterexample is surveyed, re-proved and extended to every dimension by Cristian Anghel","takeaway":"OpenAI's 6 Oct release included a model-written counterexample to the domain form of Hochster's small Cohen–Macaulay conjecture, a central question of commutative algebra since 1973. A survey by Cristian Anghel, whose September preprints started this line of work, gives a complete proof of it and generalizes it to all dimensions.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":5,"official":5,"filed":"2026-10-22","updated":"2026-10-22","orgs":["OpenAI","Simion Stoilow Institute of Mathematics of the Romanian Academy"]},{"id":"2026-10-13-thompson-group-f-nonamenable-kim-lodha-exposition","url":"https://postcutoff.com/e/2026-10-13-thompson-group-f-nonamenable-kim-lodha-exposition/","date":"2026-10-13","date_precision":"day","short_title":"Thompson's group F is nonamenable","deck":"Specialists Sang-hyun Kim and Yash Lodha publish a human-written version of OpenAI's proof","takeaway":"The amenability of Thompson's group F, open since the 1970s, was one of the headline claims of OpenAI's 6 Oct math release. Two experts on groups of piecewise linear maps have now written the proof out themselves and generalized it, without AI.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":3,"official":3,"filed":"2026-10-14","updated":"2026-10-19","orgs":["OpenAI","KIAS","Purdue University"]},{"id":"2026-10-06-openai-math-release-722-manuscripts","url":"https://postcutoff.com/e/2026-10-06-openai-math-release-722-manuscripts/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI releases 722 AI-written math manuscripts claiming hundreds of open problems","deck":"Including quasi-Riemann, Unique Games, Hodge for CM abelian varieties and free group factors","takeaway":"If even a fraction of these results hold up, this is the largest single jump in mathematical knowledge on record, produced by an AI system in about six weeks.","category":"science","category_label":"Science & math","importance":5,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":84,"official":13,"filed":"2026-10-07","updated":"2026-10-20","orgs":["OpenAI"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-22","type":"filed","text":"Created (PDF read in full text)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-22","run":null,"sources_read":"PDF read in full text","updated":"2026-10-22","human_review":null,"version":null},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":174,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":113,"in_training_data":false},{"model_id":"gemini-4-argon","name":"Gemini 4 Argon","cutoff":"2026-03","days_after":204,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":143,"in_training_data":false}],"short_url":"https://postcutoff.com/s/algebraic-geometer-jenia-tevelev"}