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Fujita's freeness conjecture
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Abstract
We prove Fujita's freeness conjecture. If X is a smooth complex projective variety of dimension n and L is an ample line bundle, then $K_X+mL$ is globally generated for every integer $m\geq n+1$.
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work in progress
Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-23) and not checked by a person. May be wrong or incomplete. Report an error.
Follow-up works
- Kenta Kitamura (KitaKen1), Add a conjecture: mixed Fujita freeness and linear effective base-point freeness (formal-conjectures issue #7012, 2026-10-09). Proposes a "mixed" freeness statement (K_X + L_1 + ... + L_s + N globally generated for ample L_i, s ≥ n+1, nef N) and a klt base-point-freeness variant, with a proposed proof sketch in an accompanying repository; neither cites this paper, and the author discloses assistance from ChatGPT and OpenAI Codex.
Comments and discussion
- Marcus Bintz (@spurious-bows.bsky.social), post on Bluesky (2026-10-12). Pairs this manuscript with the release's "The ordinary-double-point gap in every dimension" (9/24), matching the proof of Prop 3.2 to that paper's Lemma 5.2, Lemma 3.1 and Eq. 5.5 to its Prop 4.1(i), and sections 4-5 broadly to its sections 4.2-4.4.
Related prior literature mentioned by others
- Jingjun Han, A linear bound for Fujita's freeness conjecture (arXiv:2609.01574, revised 11 September 2026). The AI Proved What? exposition highlights this prior bound, $m\geq\lceil C_0n\rceil$ with $C_0=1.77629\ldots$, as a comparison with the OpenAI manuscript's claimed sharp bound $n+1$.
- Luca Ghidelli and Justin Lacini, Logarithmic bounds on Fujita's conjecture (arXiv:2107.11705, 2021; Proceedings of the London Mathematical Society, 2024). The AI Proved What? exposition places this earlier bound of order $n\log\log n$ in the progression of general-dimensional freeness bounds preceding the manuscript.
Possibly relevant literature (found by the agent, not checked)
- Antonio Laface, Fujita freeness for projectivized toric vector bundles (arXiv:2608.28438, 2026-08-28). Claims global generation of K_Y+mA in the sharp range for projectivized toric vector bundles, a special case of the paper's Theorem 1.1; the preprint discloses ChatGPT assistance.
- Klaus Altmann, Nathan Ilten, Fujita's Freeness Conjecture for T-Varieties of Complexity One (arXiv:1712.09927, 2017). Proves the freeness conjecture for Gorenstein complexity-one T-varieties with rational singularities, a special-class result alongside the dimension-by-dimension history recounted in the paper's introduction.
- Takumi Murayama, Moving Seshadri constants and effective Fujita-type conjectures (arXiv:2408.06530, v2 2026-03-23). Studies a Fujita-type variant for |K_X + aK_X + bL| in arbitrary characteristic using moving Seshadri constants, a different approach to the effective freeness questions the paper addresses.
Digests and expositions
- AI Proved What?, Sections everywhere? Fujita’s freeness conjecture (episode 038, October 2026). This video and accompanying transcript explain the manuscript's claimed bound and its counting, exponential minimization, point-center reduction, and vanishing argument, while distinguishing freeness from very ampleness. [AI-generated per source] [family-level]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 038
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Fujitas-freeness-conjecture-September-23-2026,
author = {{OpenAI}},
title = {{Fujita's freeness conjecture}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Fujitas-freeness-conjecture-September-23-2026/Fujitas-freeness-conjecture-September-23-2026.pdf}{OAI:Fujitas-freeness-conjecture-September-23-2026}},
year = {2026}
}Family 038: Fujita’s freeness conjecture
Proves Fujita's freeness conjecture at its sharp bound in every dimension: for a smooth projective complex variety X of dimension n and an ample line bundle L, the adjoint $K_X+mL$ is globally generated for every integer $m\ge n+1$.
This is the only paper in this family.