Nagata's conjecture on plane curves through very general points

Theorem 1.1: for every $r\ge10$ there is a countable union $E_r$ of proper Zariski-closed subsets of the configuration space, with nonempty complement, such that for every $(p_1,\dots,p_r)\notin E_r$ , every nonzero effective plane curve $C$ of degree $d\ge1$ and all $m_i\le\mathrm{mult}_{p_i}C$ , one has $\sum m_i<d\sqrt r$ . Reducible and nonreduced curves and unequal multiplicities are included. Section 6 deduces that the multipoint Seshadri constant of a line is $1/\sqrt r$ and extends the theorem to every uncountable algebraically closed field of characteristic zero. It does not treat positive characteristic in this manuscript, and it does not address the SHGH conjecture or other finer statements about linear systems.

Exact FrontierDelta

Prior state unknown→proved

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Occurred: Sep 23, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: Nagata's conjecture on plane curves through very general points · Nagata's conjecture

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VibeMathed record: Nagata's conjecture on plane curves through very general points

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No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the manuscript was read against Nagata's conjecture; it gives the strict, nonhomogeneous form for every $r\ge10$ , with one countable union of proper closed exceptional sets chosen simultaneously for all degrees and multiplicity vectors. The proof was not refereed. lean/formalization.yaml lists a main result for this manuscript (comparator Nagata, declaration OAI.Nagata.nagata_conjecture). The comparator statement ComparatorChallenges/Nagata.lean was read here: for every $r\ge10$ it asserts a countable family of proper Zariski-closed subsets of ordered distinct configurations in $\mathbb P^2(\mathbb C)$ , with a configuration outside all of them, such that outside them every nonzero effective plane curve (a cycle of homogeneous prime ideals, repeated components allowed) satisfies $\sum m_i<d\sqrt r$ whenever $m_i$ is at most its multiplicity at $p_i$ . This states the headline claim, over $\mathbb C$ . Not rebuilt here. Permitted axioms: propext, Quot.sound, Classical.choice.