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Math-2538 · mathematics · Algebraic geometry

Canonical minimal models in numerical dimension one

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
If a smooth connected projective complex n-fold, $n\geq 3$, has pseudo-effective $K_X$ with Nakayama numerical Kodaira dimension $\kappa _\sigma (K_X)=1$, must a finite sequence of K-negative divisorial contractions and flips reach a projective Q-factorial terminal model Y with nef $K_Y$? On a common resolution must $p*K_X=q*K_Y+E$ hold with E effective q-exceptional and containing every contracted X-prime in its support?

Short English statement

If a smooth connected projective complex n-fold, $n\geq 3$, has pseudo-effective $K_X$ with Nakayama numerical Kodaira dimension $\kappa _\sigma (K_X)=1$, must a finite sequence of K-negative divisorial contractions and flips reach a projective Q-factorial terminal model Y with nef $K_Y$? On a common resolution must $p*K_X=q*K_Y+E$ hold with E effective q-exceptional and containing every contracted X-prime in its support?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2538-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2538

Manuscript 1 (family 036, 2026-09-24): 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.

Canonical minimal models in numerical dimension one

OpenAI's claim

A smooth complex projective variety with canonical section-growth numerical dimension one is claimed to admit a terminal factorial minimal model, with good-model consequences using the separate abundance input. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
Let $X$ be smooth connected projective over $\mathbb C$, $n=\dim X\ge3$, and suppose $K_X$ is pseudo-effective. Define the section-growth numerical dimension by \[ \kappa_\sigma(X,D)=\max\{k\in\{0,\ldots,n\}:\ \exists\text{ ample Cartier }A,\ \limsup_{m\to\infty}h^0(X,mD+A)/m^k>0\}. \] Assume $\kappa_\sigma(X,K_X)=1$. OpenAI claims finitely many divisorial contractions or flips, each $K$-negative to a normal projective rational-factorial terminal $Y$ with $K_Y$ nef. The inverse map contracts no divisor, and on a common smooth resolution $p:W\to X$, $q:W\to Y$, \[ p^*K_X=q^*K_Y+E,\qquad E\ge0,\quad q_*E=0, \] with the strict transforms of all contracted prime divisors in $\operatorname{Supp}E$. A separate corollary, using the companion log-abundance theorem, claims $K_Y$ semiample. The invariant retains the fixed ample twist $A$; it is not an assumed growth bound for untwisted pluricanonical sections.

Outline of the argument

Manuscript 1 (family 036, 2026-09-24): Positive-boundary programs first reduce the possible exceptional loci. Algebraic codimension-two cycle classes give a finite homological decrease after discrepancy control; a final surface criterion eliminates the remaining obstruction without assuming general minimal-model existence.

Formal verification

Family 036 has no Lean scope page in this snapshot; no formalization of these manuscript results is listed.

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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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