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Math-2237 · mathematics · Algebraic geometry

Wikipedia geometry item 13: Nagata's conjecture on curves, specifically the minimal degree required for a plane algebraic c…

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Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
For $r\geq10$ very general points $p_1,\ldots,p_r$ in the complex projective plane, is the degree $d$ of every plane curve with assigned multiplicities $m_i$ constrained by \[d\sqrt r\geq\sum_{i=1}^rm_i?\] In its usual nontrivial formulation the inequality is strict for effective curves in the range with nonzero multiplicities.

Formulation references: [S1], [S2].

Short English statement

For $r\geq10$ very general points $p_1,\ldots,p_r$ in the complex projective plane, is the degree $d$ of every plane curve with assigned multiplicities $m_i$ constrained by \[d\sqrt r\geq\sum_{i=1}^rm_i?\] In its usual nontrivial formulation the inequality is strict for effective curves in the range with nonzero multiplicities.

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2237-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2237

This page records Nagata's degree-versus-multiplicity inequality for very general plane points. The theorem asserts a strict inequality for every point count at least ten, allowing unequal nonnegative multiplicities and nonzero reducible curves. It therefore supplies the stated non-strict bound and the strict form described here. A classification of every exact minimum degree is not part of this page's mathematical definition.

Wikipedia geometry item 13: Nagata's conjecture on curves, specifically the minimal degree required for a plane algebraic c…

OpenAI's claim

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. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
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, 2026-09-23): 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. The documented statement covers the main claim of manuscript 1. 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

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