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Math-369-A2 · revision r1 · on Math-369

OpenAI claim summary for Math-369

Literature report · LLM-generated · unknown ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status: open. Claims here do not change the source status.

Claim: literature report

Summary of a claim published in openai/math; not a claim by this platform.

The separate Nagata manuscript covers exactly the strict inequality for all point counts at least ten, with unequal assigned multiplicities and nonzero effective curves at very general complex plane points. The polarized-surface companion gives only eventual maximal constants and a non-strict curve slope bound. The higher-dimensional companion assumes dimension at least three, so it does not apply to plane curves. The solved claim here comes from Nagata's plane theorem.

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. Nagata Conjecture on Plane Curve Linear Systems
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Manuscript 1.
  5. Manuscript 2.
  6. Manuscript 3.
  7. Outline of the argument
  8. Formal verification
  9. Status caveat
  10. Sources

Nagata Conjecture on Plane Curve Linear Systems

OpenAI's claim

Manuscript 1 (family 039, 2026-10-05): In every dimension at least three, the ordinary multipoint Seshadri constant is claimed to reach $(L^n/r)^{1/n}$ at very general points for every sufficiently large $r$. Manuscript 2 (family 039, 2026-09-23): For any polarized smooth complex projective surface, the equal-weight multipoint Seshadri constant is claimed to reach $\sqrt{L^2/r}$ for all integers beyond a polarization-dependent threshold. Manuscript 3 (family 039, 2026-09-23): At $r\ge10$ very general complex plane points, every effective curve is claimed to have total assigned multiplicity below its degree times $\sqrt r$, including reducible curves and unequal assignments.

Mathematical statement of the claimed result

Manuscript 1.
Let $X$ be smooth integral projective over $\mathbb C$, with $n=\dim X\ge3$, and $L$ ample. OpenAI claims an integer $r_0(X,L)$ such that for every $r\ge r_0$ and very general ordered distinct $p_1,\ldots,p_r$, \[ \varepsilon(X,L;\mathbf p)=\left(\frac{L^n}{r}\right)^{1/n}. \] The multipoint Seshadri constant is the supremum of $t\ge0$ for which $\pi^*L-t\sum_iE_i$ is nef on the blow-up at these points. For each $r$, the exceptional tuples lie in a union, indexed by a countable set, of proper algebraic closed subsets with nonempty complement. The threshold depends on $(X,L)$ and is not specified uniformly.
Manuscript 2.
For any smooth integral projective complex surface $S$ and ample line bundle $L$, OpenAI claims $r_0(S,L)$ such that every $r\ge r_0$ and very general tuple of distinct points give, on the blow-up $\pi$ with exceptional curves $E_i$, \[ \pi^*L-\sqrt{L^2/r}\sum_{i=1}^rE_i\text{ nef},\qquad \varepsilon(S,L;\mathbf p)=\sqrt{L^2/r}. \] The constant is the supremum of $t$ making $\pi^*L-t\sum E_i$ nef. Very general excludes countably many proper closed subsets of the distinct-point configuration space; the statement is for sufficiently many points, not every prescribed small $r$.
Manuscript 3.
For each $r\ge10$, OpenAI claims a nonempty very-general locus of ordered distinct points $(p_1,\ldots,p_r)$ in $\mathbb P^2_{\mathbb C}$ such that every nonzero effective plane curve $C$ of degree $d\ge1$ and integers $m_i\ge0$ with $\operatorname{mult}_{p_i}C\ge m_i$ satisfy \[ d\sqrt r>\sum_{i=1}^r m_i. \] Very general means excluding a union, indexed by a countable set, of proper algebraic closed subsets of the distinct-point configuration space. Multiplicities may be unequal, curves may be reducible, and the strict inequality starts at ten points.

Outline of the argument

Manuscript 1 (family 039, 2026-10-05): Graded exponent supports are reshaped into a volume-preserving simplex compatible with section multiplication. Full density forces every sufficiently deep interior lattice point to occur, allowing torus interpolation to separate the needed jets; very-general tuples inherit all the finite tests.

Manuscript 2 (family 039, 2026-09-23): Finite jet tests are transferred to an algebraic torus through nodal-curve degeneration. A varying exponent region is tuned to a primitive lattice direction and a subgroup with exactly $r$ points, giving injectivity for all required orders rather than only a fixed multiplicity.

Manuscript 3 (family 039, 2026-09-23): Incidence loci and cyclic products reduce a hypothetical failure to equal multiplicities at arbitrary configurations. Degeneration toward a cubic produces finite jet constraints; colliding the remaining points and taking limits of elliptic theta bases turns the contradiction into a full-rank polynomial evaluation problem.

Formal verification

For family 039, the scope page documents the following selections. The selected plane-curve inequality allows unequal multiplicities and repeated components at every allowed point count. A separate statement gives eventual maximality on each polarized complex surface, with a surface-dependent threshold; the higher-dimensional and positive-characteristic companions are not selected here. Manuscript 1 is not among the papers selected by that page, so the family link does not formalize its main conclusion. The documented statement covers the main claim of manuscript 2. The documented statement covers the main claim of manuscript 3. Its selected declaration also covers the clause used for this problem. Comparator files: MaximalSeshadriConstants.lean, Nagata.lean. Declarations: OAI.MaximalSeshadri.Geometry.maximalSeshadriConstants, OAI.Nagata.nagata_conjecture.

This site has not executed the Lean files or checked their dependencies. These descriptions report the scope of the linked declarations, rather than a new verification of the proofs.

Status caveat

OpenAI attributes these results to an internal model. Its release notice cautions that results lacking full formalization can contain mistakes. This site has not independently assessed the proofs; a solved label records the scope of the reported claim.

Sources