Nagata's conjecture on plane curves through very general points

Let $p_1,\dots,p_r$ be very general points of the complex projective plane. Nagata, in his work on Hilbert's fourteenth problem, conjectured that for $r\ge 10$ every nonzero effective plane curve $C$ of degree $d$ with $\mathrm{mult}_{p_i}C\ge m_i$ satisfies $\sum_{i=1}^r m_i < d\sqrt r$ . He proved it when $r$ is a perfect square at least $16$ . The bound fails at $r=9$ (a cubic through nine points), and partial results (Roé, Harbourne, Ciliberto-Miranda degenerations, the $117/37$ bound at ten points) left every nonsquare $r\ge10$ open. Equivalently, the multipoint Seshadri constant of a line at $r\ge10$ very general points is $1/\sqrt r$ . Does the strict inequality $\sum_i m_i<d\sqrt r$ hold for every $r\ge 10$ , every degree and every multiplicity vector?

Result: Proved(see note)
Status: Candidate (review pending)
AI contribution: AI-discovered
Method: Argument
Field: Algebraic geometry; linear systems of plane curves
Posed by: Masayoshi Nagata
Year posed: 1959
Years open: 67y
Solved: 2026-09-23
Model: Unreleased internal OpenAI model
Vendor: OpenAI
Collaborators: —
Verification: Lean-checked, statement unaudited
Publication: Announced
Collection: OpenAI math release (October 2026), version adc7f12
Significance: 62 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

Theorem 1.1: for every $r\ge10$ there is a countable union $E_r$ of proper Zariski-closed subsets of the configuration space, with nonempty complement, such that for every $(p_1,\dots,p_r)\notin E_r$ , every nonzero effective plane curve $C$ of degree $d\ge1$ and all $m_i\le\mathrm{mult}_{p_i}C$ , one has $\sum m_i<d\sqrt r$ . Reducible and nonreduced curves and unequal multiplicities are included. Section 6 deduces that the multipoint Seshadri constant of a line is $1/\sqrt r$ and extends the theorem to every uncountable algebraically closed field of characteristic zero. It does not treat positive characteristic in this manuscript, and it does not address the SHGH conjecture or other finer statements about linear systems.

What the AI did

The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. 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). The manuscript is authored 'OpenAI' and names no human author. The principal manuscript (September 23, 2026) is self-contained; the Seshadri-constant companions in the same family cite it.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the manuscript was read against Nagata's conjecture; it gives the strict, nonhomogeneous form for every $r\ge10$ , with one countable union of proper closed exceptional sets chosen simultaneously for all degrees and multiplicity vectors. The proof was not refereed. lean/formalization.yaml lists a main result for this manuscript (comparator Nagata, declaration OAI.Nagata.nagata_conjecture). The comparator statement ComparatorChallenges/Nagata.lean was read here: for every $r\ge10$ it asserts a countable family of proper Zariski-closed subsets of ordered distinct configurations in $\mathbb P^2(\mathbb C)$ , with a configuration outside all of them, such that outside them every nonzero effective plane curve (a cycle of homogeneous prime ideals, repeated components allowed) satisfies $\sum m_i<d\sqrt r$ whenever $m_i$ is at most its multiplicity at $p_i$ . This states the headline claim, over $\mathbb C$ . Not rebuilt here. Permitted axioms: propext, Quot.sound, Classical.choice.

Nagata's conjecture on plane curves through very general points

What was actually shown

What the AI did

Verification

Sources

Discussion