Browse › Algebraic and complex geometry › 039 Nagata’s conjecture and maximal Seshadri constants
Maximal Seshadri constants on arbitrary polarized surfaces
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Abstract
We prove that, for every smooth integral complex projective surface S and every ample line bundle L, the multipoint Seshadri constant at r very general points equals $\sqrt{L^2/r}$ for every sufficiently large integer r. This resolves positively the qualitative Nagata–Biran conjecture for surfaces.
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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 eventual maximality $\varepsilon(S,L;\mathbf p)=\sqrt{L^2/r}$ at very general points for every smooth complex projective surface with ample $L$, with a non-explicit threshold; comparator statement in
ComparatorChallenges/MaximalSeshadriConstants.lean. [family-level]
Related prior literature mentioned by others
- Oliver Küchle, Multiple point Seshadri constants and the dimension of adjoint linear series (Annales de l’Institut Fourier 46(1), 1996). The family exposition mentions these asymptotically optimal bounds on complex varieties as background to eventual maximality. [family-level]
Possibly relevant literature (found by the agent, not checked)
- Grzegorz Malara, Łukasz Merta, Justyna Szpond, Marcin Zieliński, Dihedral reflections and an infinite series of irrational Seshadri constants (arXiv:2610.01783, 1 Oct 2026). Extends the Laface–Ugaglia construction to compute irrational Seshadri constants of a line bundle on $\mathbb P^1\times\mathbb P^1$ at $2n$ very general points for odd $n\ge5$; relevant to the paper's discussion of exact fixed-surface values (Dionne–Roth, Laface–Ugaglia) as opposed to eventual maximality.
- Antonio Trusiani, Multipoint Okounkov bodies (Annales de l’Institut Fourier 71(6), 2021). Its convex-body descriptions of multipoint Seshadri constants, volumes and Kähler packings provide a possible comparison for the manuscript’s claimed volume-bound equality on complex varieties. [family-level]
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:Maximal-Seshadri-Constants-on-Arbitrary-Polarized-Surfaces-September-23-2026,
author = {{OpenAI}},
title = {{Maximal Seshadri constants on arbitrary polarized surfaces}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Maximal-Seshadri-Constants-on-Arbitrary-Polarized-Surfaces-September-23-2026/main.pdf}{OAI:Maximal-Seshadri-Constants-on-Arbitrary-Polarized-Surfaces-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:
- Nagata's conjecture for plane curves · September 23, 2026
- Maximal Multipoint Seshadri Constants in Higher Dimensions · October 5, 2026
- Maximal multipoint Seshadri constants in positive characteristic · October 5, 2026