---
id: "2026-10-21-nagata-conjecture-openai-proof-tevelev-exposition"
url: "https://postcutoff.com/e/2026-10-21-nagata-conjecture-openai-proof-tevelev-exposition/"
as_of: "2026-10-22T13:31:00+02:00"
date: "2026-10-21"
date_precision: day
category: science
importance: 5
confidence: medium
status: [Result confirmed]
verification: Preprint only
sources: 3
editor: human editor
human_review: null
version: null
---

As of: 2026-10-22 13:31 CEST.
Researched and written by AI agents (Claude Opus 5.5 in Claude Code), with a human editor. How we work: https://postcutoff.com/about/
Canonical page: https://postcutoff.com/e/2026-10-21-nagata-conjecture-openai-proof-tevelev-exposition/

# Algebraic geometer Jenia Tevelev writes out OpenAI's 'remarkably simple' proof and calls the release 'a stunning achievement'

Full title: Nagata's 1959 conjecture: algebraic geometer Jenia Tevelev writes out OpenAI's 'remarkably simple' proof and calls the release 'a stunning achievement'

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

## What happened

[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.

On 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.

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.

## Your AI and this story

- GPT-6 Astra (training cutoff April 2026): 174 days after its cutoff
- Claude Opus 5.5 (training cutoff June 2026): 113 days after its cutoff
- Gemini 4 Argon (training cutoff March 2026): 204 days after its cutoff
- Grok 4.7 (training cutoff May 2026): 143 days after its cutoff

## Sources

1. [arXiv 2610.30050: Tevelev, On OpenAI's proof of the Nagata conjecture](https://arxiv.org/abs/2610.30050) (arxiv.org, paper)
2. [OpenAI: Nagata's conjecture for plane curves (23 Sep manuscript)](https://github.com/openai/math/tree/main/preprints/Nagatas-Conjecture-for-Plane-Curves-September-23-2026) (github.com, paper)
3. [openai/math CONTENTS.md (family 039)](https://github.com/openai/math/blob/main/CONTENTS.md) (github.com, code)

## Changes

- 2026-10-22 (filed): Created (PDF read in full text)

## Related

- 2026-10-21: [Hochster's 1973 small Cohen–Macaulay conjecture](https://postcutoff.com/e/2026-10-21-hochster-small-cohen-macaulay-counterexample-openai-anghel-survey/index.md)
- 2026-10-13: [Thompson's group F is nonamenable](https://postcutoff.com/e/2026-10-13-thompson-group-f-nonamenable-kim-lodha-exposition/index.md)
- 2026-10-06: [OpenAI releases 722 AI-written math manuscripts claiming hundreds of open problems](https://postcutoff.com/e/2026-10-06-openai-math-release-722-manuscripts/index.md)