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Maximal Multipoint Seshadri Constants in Higher Dimensions

OpenAI · October 5, 2026

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Abstract

Let L be an ample line bundle on a smooth integral complex projective variety X of dimension n ≥ 3. 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 r very general points equals the volume bound $(L^n/r)^{1/n}$. This establishes the qualitative Nagata–Biran–Szemberg assertion in these dimensions.

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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.

Possibly relevant literature (found by the agent, not checked)

  • 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]
  • Jaesun Shin, Extended Okounkov bodies and multi-point Seshadri constants (arXiv:1710.04351, 2017; revised 25 June 2018). Its convex-geometric treatment of multipoint local positivity may provide context for the manuscript’s use of graded exponent sets and simplex reshaping to obtain maximal Seshadri constants. [family-level]

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Cite

@misc{OAI:Maximal-Multipoint-Seshadri-Constants-in-Higher-Dimensions-October-5-2026,
  author = {{OpenAI}},
  title = {{Maximal Multipoint Seshadri Constants in Higher Dimensions}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Maximal-Multipoint-Seshadri-Constants-in-Higher-Dimensions-October-5-2026/main.pdf}{OAI:Maximal-Multipoint-Seshadri-Constants-in-Higher-Dimensions-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.

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