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Nagata's conjecture for plane curves
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Abstract
We prove that a nonzero effective plane curve of degree d at r ≥ 10 very general complex points has total multiplicity strictly less than $d\sqrt r$. The inequality holds simultaneously for all curves, including reducible and nonreduced curves, and establishes Nagata's conjecture in its strict, nonhomogeneous form.
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Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Formalization
- OpenAI, Nagata’s conjecture and maximal Seshadri constants: Lean scope note (OpenAI Math Release, lean/docs/039.md, 6 October 2026). The release's own scope note says the Lean development proves the strict inequality for all curves and multiplicity vectors outside one countable union of proper closed sets, with the comparator statement in
ComparatorChallenges/Nagata.lean. [family-level]
Related prior literature mentioned by others
- Joaquim Roé, On the existence of plane curves with prescribed multiple points (arXiv:math/9807066v2, 18 May 1999). The family exposition mentions Roé’s earlier homogeneous-multiplicity estimates as background to the claimed sharp Nagata inequality.
Possibly relevant literature (found by the agent, not checked)
- Piotr Pokora, Tomasz Szemberg, Lifting negative curves from finite fields and Seshadri constants at ten points (arXiv:2610.06225, 5 Oct 2026). Gives a new lower bound 8658/27379 for the multipoint Seshadri constant of a line at ten very general points, improving Petrakiev's 228/721. This relates to the paper's discussion of the known ten-point bounds (117/37 from Eckl and Ciliberto–Dumitrescu–Miranda–Roé) and to its claimed value 1/√10 for r = 10.
Digests and expositions
- AI Proved What?, Nagata was right? Nagata’s conjecture and maximal Seshadri constants (episode 039, October 2026). This AI-assisted exposition and public transcript summarize all four claims and outline the plane and surface proof strategies. [AI-generated per source] [family-level]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 039
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Nagatas-Conjecture-for-Plane-Curves-September-23-2026,
author = {{OpenAI}},
title = {{Nagata's conjecture for plane curves}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Nagatas-Conjecture-for-Plane-Curves-September-23-2026/main.pdf}{OAI:Nagatas-Conjecture-for-Plane-Curves-September-23-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/039.md. The Lean statements have not been checked by people against the paper's claims.
Family 039: Nagata’s conjecture and maximal Seshadri constants
Proves Nagata's strict inequality $\sum_i m_i\lt d\sqrt r$ for every nonzero effective plane curve of degree d through r ≥ 10 very general complex points, with arbitrary multiplicities mi. It also proves maximal multipoint Seshadri constants $(L^n/r)^{1/n}$ for every smooth polarized projective variety of dimension n ≥ 2 and all sufficiently large r: at very general points over ℂ, and at the geometric generic tuple over any algebraically closed field of positive characteristic.
Other papers in this family:
- Maximal Seshadri constants on arbitrary polarized surfaces · September 23, 2026
- Maximal Multipoint Seshadri Constants in Higher Dimensions · October 5, 2026
- Maximal multipoint Seshadri constants in positive characteristic · October 5, 2026