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Math-2590 · mathematics · Algebraic geometry

Terminal numerical-dimension-one abundance in positive characteristic

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For a normal projective Q-factorial terminal threefold over an algebraically closed field of characteristic $p>3$, if $K_X$ is nef with $\nu (K_X)=1$, must some positive Cartier multiple have at least two independent sections, so that $\kappa (K_X)=1$ and $K_X$ is semiample?

Short English statement

For a normal projective Q-factorial terminal threefold over an algebraically closed field of characteristic $p>3$, if $K_X$ is nef with $\nu (K_X)=1$, must some positive Cartier multiple have at least two independent sections, so that $\kappa (K_X)=1$ and $K_X$ is semiample?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2590-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2590

Manuscript 1 (family 035, 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.

Terminal numerical-dimension-one abundance in positive characteristic

OpenAI's claim

For terminal factorial projective threefolds in characteristic $p>3$, a nef canonical divisor of numerical dimension one is claimed to have Kodaira dimension one and be semiample. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
Over an algebraically closed field of characteristic $p>3$, let $X$ be a normal projective rational-factorial terminal threefold with nef $K_X$. If an ample Cartier $H$ satisfies $K_XH^2>0$ and $K_X^2H=0$, OpenAI claims \[ \exists m>0:\ mK_X\text{ Cartier},\ h^0(X,mK_X)\ge2,\qquad\kappa(X,K_X)=1, \] and $K_X$ semiample. This is the boundary-zero terminal numerical-dimension-one theorem in the stated prime range.

Outline of the argument

Manuscript 1 (family 035, 2026-09-24): The proof starts from known positive-characteristic nonvanishing and seeks a second independent canonical section. A primitive-cycle obstruction combines tangent-sheaf foliations, strongly semistable inequalities, surface abundance, and Cartier descent; the resulting positive Kodaira dimension invokes the existing semiampleness theorem.

Formal verification

Family 035 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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