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Maximal multipoint Seshadri constants in positive characteristic
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Abstract
Let L be an ample line bundle on a smooth integral projective variety X over an algebraically closed field of positive characteristic, with $\dim X=n\ge3$. We prove that there is a threshold $r_0=r_0(X,L)$ such that for every integer $r\ge r_0$, the ordinary multipoint Seshadri constant at the geometric generic tuple of r points equals the volume bound $(L^n/r)^{1/n}$. The same conclusion holds in dimension two. This establishes the positive-characteristic form of the qualitative Nagata–Biran–Szemberg assertion at geometric generic tuples.
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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)
- Jaesun Shin, Extended Okounkov bodies and multi-point Seshadri constants (arXiv:1710.04351, 2017; revised 25 June 2018). Its convex-geometric treatment of multipoint positivity is a possible comparison for the manuscript’s graded-support method, with applicability in positive characteristic left unchecked. [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
- Emergent Mind: family 039
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Maximal-multipoint-Seshadri-constants-in-positive-characteristic-October-5-2026,
author = {{OpenAI}},
title = {{Maximal multipoint Seshadri constants in positive characteristic}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Maximal-multipoint-Seshadri-constants-in-positive-characteristic-October-5-2026/seshadri-positive-characteristic.pdf}{OAI:Maximal-multipoint-Seshadri-constants-in-positive-characteristic-October-5-2026}},
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 Seshadri constants on arbitrary polarized surfaces · September 23, 2026
- Maximal Multipoint Seshadri Constants in Higher Dimensions · October 5, 2026