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Math-2593 · mathematics · Algebraic geometry

Uniform log-Iitaka systems with fixed coefficients

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For each $d\geq 5$ and finite $\Phi \subset [0,1]\cap Q$, does one $m(d,\Phi )>0$ work for every normal integral projective lc d-fold (X,B) over any algebraically closed characteristic-zero field, with nonzero B coefficients in $\Phi $, rational Cartier $D=K_X+B$ and $\kappa (D)\geq 0$, so that $H^{0}(X,O_X(\lfloor mD\rfloor ))$ is nonzero and its ratios generate the entire Iitaka field K(D) inside k(X)?

Short English statement

For each $d\geq 5$ and finite $\Phi \subset [0,1]\cap Q$, does one $m(d,\Phi )>0$ work for every normal integral projective lc d-fold (X,B) over any algebraically closed characteristic-zero field, with nonzero B coefficients in $\Phi $, rational Cartier $D=K_X+B$ and $\kappa (D)\geq 0$, so that $H^{0}(X,O_X(\lfloor mD\rfloor ))$ is nonzero and its ratios generate the entire Iitaka field K(D) inside k(X)?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2593-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2593

Manuscript 1 (family 034, 2026-10-04): The problem selects the mathematical statement summarized here, including its stated hypotheses and parameter range. The cited theorem gives a claimed proof of that assertion in full.

Uniform log-Iitaka systems with fixed coefficients

OpenAI's claim

For fixed dimension and finitely many rational boundary coefficients, a uniform rounded pluricanonical degree is claimed to recover the full log Iitaka field, with bounded moduli denominators in the stated complex bundle formulas. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
Fix $d\ge5$ and finite $\Phi\subset[0,1]\cap\mathbb Q$. For every algebraically closed characteristic-zero field and normal integral projective log canonical $d$-fold pair $(X,B)$ with effective rational boundary coefficients in $\Phi$, rational Cartier $D=K_X+B$ and $\kappa(X,D)\ge0$, OpenAI claims an integer $m(d,\Phi)$ with \[ V_m(D)=H^0(X,\mathcal O_X(\lfloor mD\rfloor))\ne0,\qquad F_m(D)=K(D)\subset k(X). \] Here $F_m(D)$ is generated by ratios of nonzero sections in degree $m$, and $K(D)$ by such ratios in all positive degrees. The equality gives the entire Iitaka field, including its finite-extension information. The degree is uniform in the pair and field.

Outline of the argument

Manuscript 1 (family 034, 2026-10-04): Good models and uniform index inputs reduce the question to relative denominator control. Galois comparison covers separate inertia actions on fiber factors, while arithmetic Stein-degree bounds and exact rounded-section comparisons preserve the full field rather than merely a generically finite image.

Formal verification

Family 034 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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