CATALOGUE 039
Nagata’s conjecture and maximal Seshadri constants
These four manuscripts concern Nagata's plane-curve inequality and ordinary, equally weighted multipoint Seshadri constants of ample line bundles. The plane paper claims a strict inequality for arbitrary multiplicities at every very general tuple with $r\geq10$. The other three claim attainment of the volume bound for every sufficiently large integer $r$ on a fixed polarized variety. Each article distinguishes the authors' claims from the editorial checking scope.
Papers in this catalogue
These summaries give the principal scope; consult the cited original results for the complete hypotheses and statements.
Page counts include references. On narrow screens, scroll the table sideways.
| # | Manuscript and overview | Principal claim | Role and available overview |
|---|---|---|---|
| 01 | Nagata's conjecture for plane curves →Short name: Nagata's conjecture for plane curves2026-09-23 · 24 pagesOriginal paper (GitHub) ↗ | For r≥10 very general complex plane points, the strict Nagata inequality for all nonzero effective curves and arbitrary multiplicities; equal-weight constant 1/√r. Theorem 1.1 · p. 1; Corollaries 6.1–6.2 · p. 22 ↗ | Specialize universal equal-multiplicity systems to a normal bundle; use coefficients for squares and collision plus theta-matrix rank otherwise. Read the proof overview → |
| 02 | Maximal Seshadri constants on arbitrary polarized surfaces →Short name: Maximal Seshadri constants on arbitrary polarized surfaces2026-09-23 · 19 pagesOriginal paper (GitHub) ↗ | For a fixed smooth integral complex projective surface and ample L, ε=√(L²/r) at very general tuples for every sufficiently large r; extended to uncountable algebraically closed characteristic-zero fields. Theorem 1.1 · p. 2; Corollary 5.5 · p. 18 ↗ | Node exponent bounds, compression and finite torus groups give injective jet tests and nefness against curves with unequal multiplicities. Read the proof overview → |
| 03 | Maximal Multipoint Seshadri Constants in Higher Dimensions →Short name: Maximal Multipoint Seshadri Constants in Higher Dimensions2026-10-05 · 20 pagesOriginal paper (GitHub) ↗ | For a fixed smooth integral complex projective variety of dimension n≥3 and ample L, ε=(L^n/r)^(1/n) at very general tuples for every sufficiently large r. Theorem 1.1 · pp. 1–2 ↗ | Iterate two-intercept reshaping, fill the interior by graded additivity, and obtain the lower bound from surjectivity onto full jets. Read the proof overview → |
| 04 | Maximal multipoint Seshadri constants in positive characteristic →Short name: Maximal multipoint Seshadri constants in positive characteristic2026-10-05 · 17 pagesOriginal paper (GitHub) ↗ | For a fixed smooth integral projective variety of dimension n≥2 with ample L over an algebraically closed field of positive characteristic, ε=(L^n/r)^(1/n) at the geometric generic tuple for all sufficiently large r, including countable fields. Theorem 1.1 · pp. 1–2; Remark 6.3 · p. 16 ↗ | Formal jet transfer, a node retaining two branches in characteristic two, and CRT adapt the method to positive characteristic. Read the proof overview → |
Methodological connections and reading routes
Dashed arrows show methodological inheritance and reconstruction explicitly described by the later manuscripts. They do not denote use of a predecessor's main theorem as a premise. The plane paper is shown separately because it gives a different argument covering every $r\geq10$. Diagram keys refer to these fixed manuscripts by OpenAI.
- OpenAI26a OpenAI — Nagata's conjecture for plane curves — A separate route to the strict plane inequality and strict nefness; 2026-09-23.
- OpenAI26b OpenAI — Maximal Seshadri constants on arbitrary polarized surfaces — Surface jet injectivity from nodes, compression and finite groups; 2026-09-23.
- OpenAI26c OpenAI — Maximal Multipoint Seshadri Constants in Higher Dimensions — Generalization to graded exponent sets and full-jet surjectivity; 2026-10-05.
- OpenAI26d OpenAI — Maximal multipoint Seshadri constants in positive characteristic — Positive-characteristic reconstruction using formal substitutions and a geometric generic tuple; 2026-10-05.
[OpenAI26a, Theorem 1.1; Corollary 6.2, pp. 1, 22]
[OpenAI26b, Theorem 1.1; Corollary 5.5, pp. 2, 18]
[OpenAI26c, Theorem 1.1, pp. 1–2]
[OpenAI26d, Theorem 1.1; Remark 6.3, pp. 1–2, 16]
In the plane paper, normal-bundle specialization and two ordered limits establish strictness, while Corollary 6.2 deduces strict nefness and the Seshadri constant. Eventual maximality on arbitrary surfaces should not be identified with strictness for every plane point count $r\geq10$.
The higher-dimensional paper reconstructs the surface node, compression and primitive-direction method as two-intercept reshaping. The final test changes from injectivity of jets on surfaces to surjectivity onto full jets in higher dimensions. The positive-characteristic paper further uses formal jet transfer and a two-branch node, imposing all tests at the geometric generic tuple. These changes accommodate characteristic two and countable algebraically closed fields.
| Methodological source | Reconstruction | Role and checked distinction |
|---|---|---|
| Surface paper, Propositions 3.2, 3.4; Lemma 4.3 | Higher-dimensional paper, §1.1; Proposition 4.1 | Remaining coordinates become labels and real intercepts are reshaped; the required arguments are reproved |
| The same three surface results and higher-dimensional §§2–6 | Positive-characteristic paper, §1.1; §§2–5 | Formal substitutions replace analytic limits; the node becomes $y^2+xy-x^3$. Interior filling and interpolation are also reproved |
[OpenAI26b, Propositions 3.2, 3.4; Lemma 4.3, pp. 8–10, 14]
Sources and checking scope
The official number and subject follow the OpenAI Research Catalog, “Algebraic and complex geometry” (032–068). Entry 039 is on overview PDF page 6. The table retains the official order in CONTENTS. The source snapshot is commit 1581d4cbdd69a0faf34b2089614fe2a814316a6a.
OpenAI Research Catalog · 039 · p. 6
The articles check every principal conclusion against the relevant proof passages and compare direct inputs with their uses. The original statement and location in Chow's paper remain unchecked in the plane article. Complete external proofs, all analytic estimates and independent verification of constructions in every degree were not undertaken. Connections with other catalogues have not been exhaustively investigated; uninvestigated does not mean independent. The diagram selects confirmed methodological relations and is not a dependency graph of every cited work.