Skip to content
DraconiticResearch
Search Sources API
203,203 studies31 sourcesUpdated 17 min ago
PreprintCC BY 4.0arXiv

On OpenAI's proof of the Nagata conjecture

  • Jenia Tevelev
Open at sourcePDFDOICite

Abstract

These notes are an expanded account of my talk at UMass Amherst on OpenAI's remarkably simple proof of the Nagata conjecture. I hope they will be useful, especially to non-experts. The standard reduction to the normal cone and the collision of fat points into one monomial scheme reduce the problem to interpolation of monomial schemes on the normal ruled surface of a smooth plane cubic. The novelty is that the study of the asymptotics of theta functions as the cubic develops a node, together with interpolation duality, reduces the problem further to interpolation on a triangle of lattice points, which is straightforward. The same calculation determines the nef and pseudoeffective cones of a weighted blow-up of Bbb P2 at one point for very general local parameters when the ratio of weights is greater than nine.

Related

  1. The geometric case of the Erdős Similarity Conjecture. A human exposition of the OpenAI proof
    Alex Iosevich, Alexia Yavicoli
    arXivPreprintarXiv licenceFull textAnalysis & PDEsProbabilityarXiv
  2. Nagata's conjecture on a polynomial automorphism in positive characteristic
    Shigeru Kuroda
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv
  3. A Tropical Viewpoint on the Nagata--Biran--Szemberg Conjecture
    Juliette Bruce
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv
  4. The Deligne--Drinfeld conjecture after OpenAI
    Thomas Willwacher
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv
  5. A Note on the OpenAI Construction of Counterexamples to the Kaplansky Direct-Finiteness Conjecture
    Andrei Jaikin-Zapirain
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryarXiv

Cite

BibTeXRIS
BibTeX
@misc{tevelev2026openais,
  title = {{On OpenAI's proof of the Nagata conjecture}},
  author = {Jenia Tevelev},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.30050},
  eprint = {2610.30050},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.30050},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - On OpenAI's proof of the Nagata conjecture
AU  - Jenia Tevelev
PY  - 2026
DA  - 2026/10/21
PB  - arXiv
DO  - 10.48550/arxiv.2610.30050
UR  - https://arxiv.org/abs/2610.30050
AB  - These notes are an expanded account of my talk at UMass Amherst on OpenAI's remarkably simple proof of the Nagata conjecture. I hope they will be useful, especially to non-experts. The standard reduction to the normal cone and the collision of fat points into one monomial scheme reduce the problem to interpolation of monomial schemes on the normal ruled surface of a smooth plane cubic. The novelty is that the study of the asymptotics of theta functions as the cubic develops a node, together with interpolation duality, reduces the problem further to interpolation on a triangle of lattice points, which is straightforward. The same calculation determines the nef and pseudoeffective cones of a weighted blow-up of Bbb P^2 at one point for very general local parameters when the ratio of weights is greater than nine.
ER  -