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Math-367-A2 · revision r1 · on Math-367

OpenAI claim summary for Math-367

Literature report · LLM-generated · unknown ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status: open. Claims here do not change the source status.

Claim: literature report

Summary of a claim published in openai/math; not a claim by this platform.

Manuscript 1 (family 038, 2026-09-23): The main theorem gives global generation of exactly K_X+(n+1)L for arbitrary smooth complex projective varieties and arbitrary ample line bundles, with no very-ampleness assumption.

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. Fujita Freeness Conjecture
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Mathematical claim 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

Fujita Freeness Conjecture

OpenAI's claim

For a smooth complex projective $n$-fold and ample $L$, the adjoint $K_X+mL$ is claimed to be generated by sections whenever $m\ge n+1$. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
For every smooth connected projective complex variety $X$ of dimension $n\ge1$ and every ample line bundle $L$, OpenAI claims \[ \mathcal O_X(K_X)\otimes L^{\otimes(n+1)}\text{ is globally generated}. \] The ample bundle need not itself have sections. This is the freeness assertion; the main theorem does not assert that $K_X+(n+2)L$ is very ample.

Outline of the argument

Manuscript 1 (family 038, 2026-09-23): An ideal on a product with a flag variety makes the minimum of normalized section orders attainable. Tangential variation at a positive-dimensional center gives an upper moment bound, while uniform restriction ranks and Brunn--Minkowski give an incompatible lower bound; the remaining point center permits vanishing-based section lifting.

Formal verification

Family 038 has no Lean scope page in this snapshot; no formalization of these manuscript results is listed.

This site has not executed the Lean files or checked their dependencies. These descriptions report the scope of the linked declarations, rather than a new verification of the proofs.

Status caveat

OpenAI attributes these results to an internal model. Its release notice cautions that results lacking full formalization can contain mistakes. This site has not independently assessed the proofs; a solved label records the scope of the reported claim.

Sources