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Math-2592 · mathematics · Algebraic geometry

Uniform effective log-Iitaka systems for fourfolds

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Given finite $\Phi \subset [0,1]\cap Q$, does some positive $m(\Phi )$ make $|\lfloor m(K_X+\Delta )\rfloor |$ nonempty and make its ratios generate the full Iitaka field for every normal integral projective complex lc fourfold $(X,\Delta )$ with effective rational boundary coefficients in $\Phi $ and rational Cartier pseudo-effective $K_X+\Delta $? The same m must work for every chosen canonical divisor.

Short English statement

Given finite $\Phi \subset [0,1]\cap Q$, does some positive $m(\Phi )$ make $|\lfloor m(K_X+\Delta )\rfloor |$ nonempty and make its ratios generate the full Iitaka field for every normal integral projective complex lc fourfold $(X,\Delta )$ with effective rational boundary coefficients in $\Phi $ and rational Cartier pseudo-effective $K_X+\Delta $? The same m must work for every chosen canonical divisor.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2592-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2592

Manuscript 1 (family 034, 2026-09-26): 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 effective log-Iitaka systems for fourfolds

OpenAI's claim

For log-canonical fourfolds with coefficients from a fixed finite rational set, a uniform rounded adjoint system is claimed to generate the full Iitaka field; a uniform klt Calabi--Yau canonical index is also claimed. 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 a finite rational set $\Phi\subset[0,1]$. OpenAI claims an integer $m=m(\Phi)>0$ working for every normal integral projective complex fourfold $X$ with effective rational $\Delta$ whose nonzero coefficients lie in $\Phi$, log canonical pair $(X,\Delta)$, and pseudo-effective rational Cartier $D=K_X+\Delta$. Namely, \[ |\lfloor mD\rfloor|\ne\varnothing,\qquad \mathbb C\left(s/t:\ 0\ne s,t\in H^0(X,\mathcal O_X(\lfloor mD\rfloor))\right)=K(D)\subset\mathbb C(X). \] Here $K(D)$ is the field generated by ratios from all positive reflexive systems $H^0(X,\mathcal O_X(\lfloor jD\rfloor))$. Equality of fields asserts the Iitaka fibration itself, rather than only the dimension of its image. The integer is independent of the chosen canonical divisor.

Outline of the argument

Manuscript 1 (family 034, 2026-09-26): Marked curve chains control degrees of rational quotient leaves with compatibility under projections and finite quotients. Scalar bounds and Frobenius estimates handle the index step; a semiample contraction and the relative big-base theorem then bring uniform floor systems back to the original pair.

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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