Result 039, Algebraic and complex geometry

Nagata’s conjecture and maximal Seshadri constants

Proves Nagata's strict inequality ∑imi<dr\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 (Ln/r)1/n(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.

Lean formalization Proof

The bigger picture

Why it matters

How much can an algebraic curve concentrate at many points before its degree must increase? These unreviewed manuscripts claim sharp limits on that concentration, alongside maximal values for a broader geometric measure of local positivity.

What changes?

At r at least 10 very general complex points, the manuscripts claim every nonzero effective degree-d plane curve has summed multiplicities strictly below d times the square root of r, including reducible curves and repeated components. Multiplicity measures vanishing at a point and may differ between points. They also report maximal multipoint Seshadri constants for every smooth integral projective variety X of dimension n at least two with ample line bundle L, for all r beyond a threshold depending on (X,L).

What does that help mathematicians do?

A multipoint Seshadri constant measures how much positivity a line bundle supplies equally at several points. The reported value is the nth root of the quotient of L's volume by r, the volume bound. This holds at very general complex points, and at geometric generic tuples over algebraically closed fields of positive characteristic. Above the stated threshold, researchers could therefore rule out curve obstructions that would force a smaller constant, rather than merely estimate how close it comes to maximality.

Are there practical applications?

The immediate value is foundational, particularly for questions about which curves can satisfy prescribed vanishing conditions. The strict plane inequality would rule out a curve whenever its requested total multiplicity reaches the stated bound, even with unequal multiplicities. The broader claims identify when volume alone determines the Seshadri constant, without asserting the same behavior at arbitrary point configurations.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

4 manuscripts

Nagata's conjecture for plane curves

September 23, 2026 24 pages Main result formalized in Lean

We prove that a nonzero effective plane curve of degree d at r ≥ 10 very general complex points has total multiplicity strictly less than drd\sqrt r. The inequality holds simultaneously for all curves, including reducible and nonreduced curves, and establishes Nagata's conjecture in its strict, nonhomogeneous form.

PDF LaTeX source
Cite (BibTeX) 
@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}
}

Maximal Seshadri constants on arbitrary polarized surfaces

September 23, 2026 19 pages Main result formalized in Lean

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 L2/r\sqrt{L^2/r} for every sufficiently large integer r. This resolves positively the qualitative Nagata–Biran conjecture for surfaces.

PDF LaTeX source
Cite (BibTeX) 
@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}
}

Maximal Multipoint Seshadri Constants in Higher Dimensions

October 5, 2026 20 pages

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 r0=r0(X,L)r_0=r_0(X,L) such that for every integer r≥r0r\ge r_0, the ordinary multipoint Seshadri constant at r very general points equals the volume bound (Ln/r)1/n(L^n/r)^{1/n}. This establishes the qualitative Nagata–Biran–Szemberg assertion in these dimensions.

PDF LaTeX source
Cite (BibTeX) 
@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}
}

Maximal multipoint Seshadri constants in positive characteristic

October 5, 2026 17 pages

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≥3\dim X=n\ge3. We prove that there is a threshold r0=r0(X,L)r_0=r_0(X,L) such that for every integer r≥r0r\ge r_0, the ordinary multipoint Seshadri constant at the geometric generic tuple of r points equals the volume bound (Ln/r)1/n(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.

PDF LaTeX source
Cite (BibTeX) 
@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's note on what the formalization covers, from lean/docs/039.md.

Nagata’s conjecture and maximal Seshadri constants

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

Nagata's conjecture asserts that a nonzero effective plane curve of degree dd, with multiplicities at least mim_i at r≥10r\ge10 very general points, satisfies ∑imi<dr\sum_i m_i<d\sqrt r.

The formalization establishes this inequality simultaneously for all curves and multiplicity vectors outside one countable union of proper Zariski-closed exceptional sets with nonempty complement. Reducible curves, repeated components, and unequal multiplicities are included, in both effective-curve and homogeneous-polynomial formulations. The formalization also includes the passage from homogeneous effective cycles to equations.

The maximality question asks whether the multipoint Seshadri constant reaches the upper bound L2/r\sqrt{L^2/r} at very general points once rr is sufficiently large. The formalized result establishes this for every smooth integral complex projective surface SS and ample line bundle LL.

For every rr above a threshold depending on (S,L)(S,L), it gives a countable union of proper closed exceptional sets with nonempty complement. Outside it, the constant is L2/r\sqrt{L^2/r} and the boundary class defining this Seshadri constant on the point blowup is nef. The threshold is not explicit or uniform over all surfaces.

Comparator links

Result Comparator statement
Nagata's conjecture Nagata.lean
Eventual maximal Seshadri constants MaximalSeshadriConstants.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 9 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.