{"id":"openai-math/Nagatas-Conjecture-for-Plane-Curves","pdf":"https://github.com/openai/math/blob/main/preprints/Nagatas-Conjecture-for-Plane-Curves-September-23-2026/main.pdf","url":"https://github.com/openai/math/tree/main/preprints/Nagatas-Conjecture-for-Plane-Curves-September-23-2026","tags":["algebraic-geometry","plane-curves","nagata-conjecture"],"score":0,"title":"Nagata's conjecture for plane curves","domain":"Algebraic and complex geometry","family":"039","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.AG","summary":"We prove that a nonzero effective plane curve of degree d at r ≥ 10 very general complex points has total multiplicity strictly less than $`d\\sqrt r`$. The inequality holds simultaneously for all curves, including reducible and nonreduced curves, and establishes Nagata's conjecture in its strict, nonhomogeneous form.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":1,"published":"2026-09-23","reactions":{},"discussion":[{"url":"https://arxiv.org/abs/2610.30050","date":"2026-10-21","kind":"web","site":"arXiv.org","text":"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…","added":"2026-10-24","title":"On OpenAI's proof of the Nagata conjecture","author":"Jenia Tevelev","record":"arxiv/2610.30050"}],"formalized":true,"family_title":"Nagata’s conjecture and maximal Seshadri constants"}