--- title: "Nagata’s conjecture and maximal Seshadri constants" authors: - "OpenAI (internal model)" year: 2026 venue: "openai/math manuscript collection (GitHub), result family 039; unreviewed" arxiv: null doi: null source: "https://github.com/openai/math/blob/main/preprints/Nagatas-Conjecture-for-Plane-Curves-September-23-2026/main.pdf" topics: - "openai-math" - "ai-generated-math" - "algebraic-and-complex-geometry" - "lean4" - "formalized" seed_rank: 1899 seed_batch: "openai-math-2026-10-07" reviewed: "2026-10-07" pool: "maths-foundations" relevance_score: 8 lineage: ai-mathematical-reasoning cites: - title: "Nagata's conjecture for plane curves" url: "https://github.com/openai/math/blob/main/preprints/Nagatas-Conjecture-for-Plane-Curves-September-23-2026/main.pdf" year: 2026 arxiv: null doi: null - title: "Maximal Seshadri constants on arbitrary polarized surfaces" url: "https://github.com/openai/math/blob/main/preprints/Maximal-Seshadri-Constants-on-Arbitrary-Polarized-Surfaces-September-23-2026/main.pdf" year: 2026 arxiv: null doi: null - title: "Maximal Multipoint Seshadri Constants in Higher Dimensions" url: "https://github.com/openai/math/blob/main/preprints/Maximal-Multipoint-Seshadri-Constants-in-Higher-Dimensions-October-5-2026/main.pdf" year: 2026 arxiv: null doi: null - title: "Maximal multipoint Seshadri constants in positive characteristic" url: "https://github.com/openai/math/blob/main/preprints/Maximal-multipoint-Seshadri-constants-in-positive-characteristic-October-5-2026/seshadri-positive-characteristic.pdf" year: 2026 arxiv: null doi: null --- # Nagata’s conjecture and maximal Seshadri constants ## One-sentence takeaway OpenAI's result family 039 (Algebraic and complex geometry) claims: Proves Nagata's strict inequality $\sum_i m_i\lt d\sqrt r$ for every nonzero effective plane curve of degree d through r ≥ 10 very general complex points, with arbitrary multiplicities mi. ## Why it matters here Pure mathematics with no direct engine use; it is in the library as part of the openai/math corpus, which records what an AI research model claims to have proved. More broadly, every card in this batch is evidence for the `ai-mathematical-reasoning` thread: per the repo README, an unreleased internal OpenAI model produced it, using about three hours of ChatGPT Pro thinking compute per result on average, across roughly 4,000 posed problems. The Lean development makes the claim machine-checkable, which is the pattern Broadside cares about for agentic verification loops. ## Key ideas - Family summary (repo): It also proves maximal multipoint Seshadri constants $(L^n/r)^{1/n}$ for every smooth polarized projective variety of dimension n ≥ 2 and all sufficiently large r: at very general points over ℂ, and at the geometric generic tuple over any algebraically closed field of positive characteristic. - *Nagata's conjecture for plane curves*: 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. - *Maximal Seshadri constants on arbitrary polarized surfaces*: We prove that, for every smooth integral complex projective surface S and every ample line bundle L, the multipoint Seshadri constant at r very general points equals $\sqrt{L^2/r}$ for every sufficiently large integer r. This resolves positively the qualitative Nagata–Biran conjecture for surfaces. - *Maximal Multipoint Seshadri Constants in Higher Dimensions*: Let L be an ample line bundle on a smooth integral complex projective variety X of dimension n ≥ 3. We prove that there is a threshold $r_0=r_0(X,L)$ such that for every integer $r\ge r_0$, the ordinary multipoint Seshadri constant at r very general points equals the volume bound $(L^n/r)^{1/n}$. This establishes the qualitative Nagata–Biran–Szemberg assertion in these dimensions. - *Maximal multipoint Seshadri constants in positive characteristic*: Let L be an ample line bundle on a smooth integral projective variety X over an algebraically closed field of positive characteristic, with $\dim X=n\ge3$. We prove that there is a threshold $r_0=r_0(X,L)$ such that for every integer $r\ge r_0$, the ordinary multipoint Seshadri constant at the geometric generic tuple of r points equals the volume bound $(L^n/r)^{1/n}$. The same conclusion holds in dimension two. - Lean scope (lean/docs/039.md): Nagata's conjecture asserts that a nonzero effective plane curve of degree $d$, with multiplicities at least $m_i$ at $r\ge10$ very general points, satisfies $\sum_i m_i