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Math-2461 · mathematics · Algebraic geometry

Fujita's Conjecture for Effective Global Generation

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Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
For an ample line bundle $L$ on a projective variety $X$, what tensor power $L^{\otimes\ell}$ is guaranteed to be globally generated or very ample? In particular, does Fujita's conjecture hold for a smooth $n$-dimensional projective $X$, with $\omega_X\otimes L^{\otimes\ell}$ globally generated for $\ell\geq n+1$ and very ample for $\ell\geq n+2$?

Formulation references: [S1], [S2].

Short English statement

For an ample line bundle $L$ on a projective variety $X$, what tensor power $L^{\otimes\ell}$ is guaranteed to be globally generated or very ample? In particular, does Fujita's conjecture hold for a smooth $n$-dimensional projective $X$, with $\omega_X\otimes L^{\otimes\ell}$ globally generated for $\ell\geq n+1$ and very ample for $\ell\geq n+2$?

Sources

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

Math-2461-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2461

Manuscript 1 (family 038, 2026-09-23): The main theorem proves the global-generation branch K_X+(n+1)L. Both precise Fujita records also require very ampleness of K_X+(n+2)L, and the Wikipedia entry does not restrict its named Fujita conjecture to the free branch; none receives a solved label for the entire bundle.

Fujita's Conjecture for Effective Global Generation

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

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