View source for Nagata's conjecture on plane curves

'''Nagata's conjecture on plane curves''' predicts that a nonzero plane curve of degree ''d'' with multiplicities ''m''<sub>1</sub>, …, ''m''<sub>''r''</sub> at ''r'' ≥ 10 very general points of the complex projective plane satisfies ''m''<sub>1</sub> + … + ''m''<sub>''r''</sub> < ''d''√''r''. Equivalently, the multipoint Seshadri constant of a line at such points equals 1/√''r''. Its analogue for an arbitrary polarized surface, the qualitative Nagata–Biran–Szemberg conjecture, asks for the maximal multipoint Seshadri constant once the number of points is large. Four manuscripts in OpenAI's public mathematics collection ([https://github.com/openai/math github.com/openai/math]), dated 23 September and 5 October 2026, claim Nagata's conjecture for every ''r'' ≥ 10 with arbitrary multiplicities, the maximal multipoint Seshadri constants on every polarized surface for all sufficiently large ''r'', and analogues in higher dimensions and in positive characteristic. The collection includes a Lean formalization of the plane conjecture and of the surface theorem. This page summarizes the published claims and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper, code or a formal proof is not presented here as independent verification.

== Background ==
* As recounted in the introductions of the September manuscripts:
** Nagata introduced the problem in his work on Hilbert's fourteenth problem and proved the case of a square number of points, ''r'' ≥ 16. At nine points a nonzero cubic always exists, so the strict inequality cannot start below ten.
** Partial results include Roé's uniform estimate for ''r'' ≥ 10, Harbourne's sharpened rational bounds, and the bound ''d''/''m'' ≥ 117/37 at ten very general points obtained by the degeneration methods of Eckl and of Ciliberto, Dumitrescu, Miranda and Roé. Ciliberto, Miranda and Roé constructed irrational square-zero nef rays on blowups at ''r'' ≥ 10 very general points, which do not determine the equal-weight Seshadri constant.
** For surfaces, the qualitative conjecture is stated by Syzdek and Szemberg, with a stronger threshold prediction by Strycharz-Szemberg and Szemberg. Biran's packing stability and later symplectic results, Harbourne's square cases and the transfer results of Roé and of Roé and Ross are the cited antecedents. Dionne and Roth computed multipoint constants for families of polarizations on '''P'''<sup>1</sup> × '''P'''<sup>1</sup>, and a preprint of Laface and Ugaglia ([https://arxiv.org/abs/2609.26521 arXiv:2609.26521]) claims the value 1/√5 for ''O''(1,1) at ten very general points of that surface.

== The original results ==
* '''Claim.''' Theorem 1.1 of ''Nagata's conjecture for plane curves'' states that for every ''r'' ≥ 10 there is a countable union of proper Zariski-closed subsets of the configuration space, with nonempty complement, outside which every nonzero effective plane curve of degree ''d'' with multiplicities at least ''m''<sub>''i''</sub> at the ''r'' points satisfies ''d''√''r'' > Σ ''m''<sub>''i''</sub>. Theorem 1.1 of ''Maximal Seshadri constants on arbitrary polarized surfaces'' states that for every smooth integral complex projective surface ''S'' and ample line bundle ''L'' there is ''r''<sub>0</sub>(''S'', ''L'') such that for every ''r'' ≥ ''r''<sub>0</sub> the multipoint Seshadri constant at very general points equals √(''L''<sup>2</sup>/''r''). The two October manuscripts state the value (''L''<sup>''n''</sup>/''r'')<sup>1/''n''</sup> for every smooth polarized projective variety of dimension ''n'' ≥ 3 at very general complex points, and in positive characteristic at the geometric generic tuple.
* ''Method, as described:'' for the plane, a failure at very general points is turned into an equal-multiplicity system existing at every configuration, using Nagata's cyclic-product reduction. Moving the points toward a smooth curve gives a polynomial on its normal bundle, which excludes the square cases. For nonsquare ''r'', the points are moved to a cubic and the remaining points collide along a line, and an interpolation rank argument on an exponent polygon, with explicit theta bases on the elliptic curves, gives the contradiction. The argument builds on the degenerations of Ciliberto and Miranda, Evain's infinitesimal limits and Dumnicki's exponent diagrams.
* '''Sources.''' The manuscripts [https://github.com/openai/math/tree/1581d4cbdd69a0faf34b2089614fe2a814316a6a/preprints/Nagatas-Conjecture-for-Plane-Curves-September-23-2026 Nagata's conjecture for plane curves], [https://github.com/openai/math/tree/1581d4cbdd69a0faf34b2089614fe2a814316a6a/preprints/Maximal-Seshadri-Constants-on-Arbitrary-Polarized-Surfaces-September-23-2026 Maximal Seshadri constants on arbitrary polarized surfaces], [https://github.com/openai/math/tree/1581d4cbdd69a0faf34b2089614fe2a814316a6a/preprints/Maximal-Multipoint-Seshadri-Constants-in-Higher-Dimensions-October-5-2026 Maximal Multipoint Seshadri Constants in Higher Dimensions] and [https://github.com/openai/math/tree/1581d4cbdd69a0faf34b2089614fe2a814316a6a/preprints/Maximal-multipoint-Seshadri-constants-in-positive-characteristic-October-5-2026 Maximal multipoint Seshadri constants in positive characteristic] at commit <code>1581d4c</code>.
* '''Formalization, as described by the repository.''' The [https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/039.md scope note] lists the two September manuscripts. It states that the formalization proves Nagata's inequality for all curves and multiplicity vectors outside one countable union of proper closed exceptional sets, including reducible curves and unequal multiplicities, and the eventual maximal Seshadri constant for every smooth integral complex projective surface and ample line bundle, with a threshold that is not explicit. The Comparator statements are <code>Nagata.lean</code> and <code>MaximalSeshadriConstants.lean</code>.

== Public follow-up work ==
{| class="wikitable"
! Date !! Work !! Relation !! Public claim !! Public status (checked 20 October 2026)
|-
| 2026-10-05 || P. Pokora and T. Szemberg, ''Lifting negative curves from finite fields and Seshadri constants at ten points'', [https://arxiv.org/abs/2610.06225v1 arXiv:2610.06225v1] || Parallel work (partial result) || Constructs, by finite-field computations over '''F'''<sub>127</sub> and a mixed-characteristic lifting and smoothing criterion, the nef class 27379''H'' − 8658(''E''<sub>1</sub> + … + ''E''<sub>10</sub>) at ten very general points, so that their multipoint Seshadri constant is at least 8658/27379, improving Petrakiev's bound 228/721. The value 1/√10 predicted by Nagata's conjecture lies just above this bound. Version 1 does not mention the OpenAI manuscripts. || arXiv preprint, version 1 of 5 October 2026.
|-
| 2026-10-08 || [https://github.com/mathvet/mathvet MathVet] (GitHub; review of openai/math, tables at repository commit <code>c76f1e4</code>, 14 October 2026) || Review (fidelity of the formal statements) || Rates this family "partial": Nagata's conjecture for ''r'' ≥ 10 with arbitrary multiplicities is fully stated, while the maximal multipoint Seshadri constants are stated only for smooth complex projective surfaces, not in dimension ''n'' ≥ 3 or in positive characteristic. It notes that the Seshadri side uses hand-built scheme-theoretic definitions and that the curve equation is chosen through an existence theorem that is itself among the checked theorems. It records sandboxed Comparator passes for <code>Nagata</code> and <code>MaximalSeshadriConstants</code>. || MathVet states that it does not claim any theorem false and reports what was machine-checked as stated.
|-
| 2026-10-08 || [https://github.com/atimics/math2/blob/2c0f00528a8bb375d02bd48a513062254a53c285/docs/COROLLARIES.md atimics/math2], entry GE-11 of <code>docs/COROLLARIES.md</code> (GitHub, commit <code>2c0f005</code>) || Uses (Lean specialization) || Lean statements of the equal-multiplicity cases for ''r'' ≥ 10 very general points: ''m''√''r'' < ''d'', ''rm''<sup>2</sup> < ''d''<sup>2</sup>, ⌊√''r''⌋''m'' < ''d'', and ''d'' ≥ 3''m'' + 1 for ''r'' = 10. The entry calls them "not new mathematics", since the paper reduces to equal multiplicities itself, and states that they use only the comparator's <code>FullNagata</code>, "because its <code>nagata_conjecture</code> depends on a sorry'd lemma". || Catalogue of conditional corollaries in a public repository; AI-assisted.
|-
| 2026-10-10 || cert-machine, openai/math lane K, run 38051470892: [https://github.com/carlostoledo1891/cert-machine/blob/3e3937cd5e4a4cacd451447c4ed8033eb0340521/corpus/openai-math/kernel-runs/38051470892/Nagata.json Nagata] and [https://github.com/carlostoledo1891/cert-machine/blob/3e3937cd5e4a4cacd451447c4ed8033eb0340521/corpus/openai-math/kernel-runs/38051470892/MaximalSeshadriConstants.json MaximalSeshadriConstants] || Verification (two kernels) || Records that Lean's default kernel and the nanoda kernel accept both solutions, in 397 s and 1,314 s. The project reports two-kernel certification of 398 of 416 challenges of the collection. || Repository.
|-
| 2026-10-10 to 2026-10-14 || [https://github.com/leanprover/con-leche/blob/90577d9e49ee3aed45da3d421bdf5dc780192304/DESIGN.md con-leche development journal] (task #348) || Verification (second kernel) || In its run over all Comparator challenges of the collection at commit <code>fd4aeeb</code>, the journal's table of exceptions (time-outs, memory, errors) lists neither challenge, so both were accepted by con-leche and by Lean's official kernel. || Development journal.
|-
| 2026-10-16
| J. Jardine, Lattice24, referee reviews [https://github.com/JJardine919/openai-math-audit/blob/a06a8c6e77974d4c6021338a55a8e91b634f67f7/reviews/rank_332.json rank_332] (higher-dimensional manuscript) and [https://github.com/JJardine919/openai-math-audit/blob/a06a8c6e77974d4c6021338a55a8e91b634f67f7/reviews/rank_440.json rank_440] (positive-characteristic manuscript) in [https://github.com/JJardine919/openai-math-audit JJardine919/openai-math-audit] (archived at [https://doi.org/10.5281/zenodo.23422995 Zenodo 10.5281/zenodo.23422995])
| Review (AI-assisted audit)
| Reports no gap in either manuscript. Both reviews add one minor note that is stated not to be a located gap: the planar reshaping moves used in the proofs would, applied to the projective plane, give the Seshadri constant 1/√10 at ten points, that is Nagata's conjecture for ten points. The reviewers therefore ask for independent expert checking of the jet-transfer-by-degeneration steps. The positive-characteristic review reports a search over these moves that stays consistent with the known values for two to eight points. Neither review audits the September manuscripts.
| Part of an audit of the collection's manuscripts that the audit describes as complete. The reviews are published under CC BY 4.0.
|}

== Notes ==
* The earlier results are due to the authors named in the Background section, as recounted by the manuscripts.
* Malara, Merta, Szpond and Zieliński ([https://arxiv.org/abs/2610.01783 arXiv:2610.01783], 1 October 2026) extend the method of Laface and Ugaglia to an infinite series of maximal multipoint Seshadri constants on '''P'''<sup>1</sup> × '''P'''<sup>1</sup> and of irrational one-point Seshadri constants; the preprint does not mention the OpenAI manuscripts.
* The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.

[[Category:OpenAI results]]